2 Notation

We use the asymptotic notation $X\ll Y$ or $X=O(Y)$ to denote $|X|\leqslant CY$ for some constant $C$ . Given an asymptotic parameter $N$ going to infinity, we use $X=o(Y)$ to denote the bound $|X|\leqslant c(N)Y$ for some function $c(N)$ of $N$ that goes to zero as $N$ goes to infinity. We also write $X\asymp Y$ for $X\ll Y\ll X$ . If we need the implied constant $C$ or decay function $c(\,)$ to depend on an additional parameter, we indicate this by subscripts, e.g. $X=o_{k}(Y)$ denotes the bound $|X|\leqslant c_{k}(N)Y$ for a function $c_{k}(N)$ that goes to zero as $N\rightarrow \infty$ for any fixed choice of $k$ .

We will frequently use probabilistic notation, and adopt the convention that boldface variables such as $\mathbf{a}$ or $\mathbf{r}$ represent random variables, whereas non-boldface variables such as $a$ and $r$ represent deterministic variables (or constants). We write $\mathbb{P}(E)$ for the probability of a random event $E$ , and $\mathbb{E}\mathbf{X}$ and $\operatorname{Var}\mathbf{X}$ for the expectation and variance of a real or complex random variable $\mathbf{X}$ ; we also use $\mathbb{E}(\mathbf{X}|E)=\mathbb{E}\mathbf{X}1_{E}/\mathbb{P}(E)$ for the conditional expectation of $\mathbf{X}$ relative to an event $E$ of non-zero probability, where of course $1_{E}$ denotes the indicator variable of $E$ . In this paper, the random variables $\mathbf{X}$ of which we will compute expectations of will be discrete, in the sense that they take only finitely many values, so there will be no issues of measurability. The essential range of a discrete random variable $\mathbf{X}$ is the set of all values $X$ for which $\mathbb{P}(\mathbf{X}=X)$ is non-zero.

By a slight abuse of notation, we also retain the traditional (in additive combinatorics) use for $\mathbb{E}$ as an average, thus $\mathbb{E}_{a\in A}f(a):=(1/|A|)\sum _{a\in A}f(a)$ for any finite non-empty set $A$ and function $f:A\rightarrow \mathbb{C}$ , where we use $|A|$ to denote the cardinality of $A$ . Thus for instance $\mathbb{E}_{a\in A}f(a)=\mathbb{E}f(\mathbf{a})$ if $\mathbf{a}$ is drawn uniformly at random from $A$ .

A function $f:A\rightarrow \mathbb{C}$ is said to be $1$ -bounded if one has $|f(a)|\leqslant 1$ for all $a\in A$ . We will frequently rely on the following probabilistic form of the Cauchy–Schwarz inequality, the proof of which is an exercise.

Lemma 2.1 (Cauchy–Schwarz).

Let $A,B$ be sets, let $f:A\rightarrow \mathbb{C}$ be a $1$ -bounded function, and let $g:A\times B\rightarrow \mathbb{C}$ be another function. Let $\mathbf{a},\mathbf{b},\mathbf{b}^{\prime }$ be discrete random variables in $A,B,B^{\prime }$ respectively, such that $\mathbf{b}^{\prime }$ is a conditionally independent copy of $\mathbf{b}$ relative to $\mathbf{a}$ , that is to say that

$$\begin{eqnarray}\mathbb{P}(\mathbf{b}=b,\mathbf{b}^{\prime }=b^{\prime }|\mathbf{a}=a)=\mathbb{P}(\mathbf{b}=b|\mathbf{a}=a)\mathbb{P}(\mathbf{b}=b^{\prime }|\mathbf{a}=a)\end{eqnarray}$$

for all $a$ in the essential range of $\mathbf{a}$ and all $b,b^{\prime }\in B$ . Then we have

(2.1) $$\begin{eqnarray}|\mathbb{E}f(\mathbf{a})g(\mathbf{a},\mathbf{b})|^{2}\leqslant \mathbb{E}g(\mathbf{a},\mathbf{b})\overline{g(\mathbf{a},\mathbf{b}^{\prime })}.\end{eqnarray}$$

We will think of this lemma as allowing one to eliminate a factor $f(\mathbf{a})$ from a lower bound of the form $|\mathbb{E}f(\mathbf{a})g(\mathbf{a},\mathbf{b})|\geqslant \unicode[STIX]{x1D702}$ , at the cost of duplicating the factor $g$ , and worsening the lower bound from $\unicode[STIX]{x1D702}$ to $\unicode[STIX]{x1D702}^{2}$ .

We also have the following variant of Lemma 2.1.

Proof. If we set $\unicode[STIX]{x1D6FA}:=\{a\in A:f(a)\geqslant \unicode[STIX]{x1D702}/2\}$ , then

$$\begin{eqnarray}f(\mathbf{a})\leqslant \frac{\unicode[STIX]{x1D702}}{2}+C1_{\mathbf{a}\in \unicode[STIX]{x1D6FA}}\end{eqnarray}$$

and hence on taking expectations

$$\begin{eqnarray}\mathbb{E}f(\mathbf{a})\leqslant \frac{\unicode[STIX]{x1D702}}{2}+C\mathbb{P}(\mathbf{a}\in \unicode[STIX]{x1D6FA}).\end{eqnarray}$$

This implies that

$$\begin{eqnarray}\mathbb{P}(\mathbf{a}\in \unicode[STIX]{x1D6FA})\geqslant \unicode[STIX]{x1D702}/2C\end{eqnarray}$$

giving the claim. ◻

If $\unicode[STIX]{x1D703}\in \mathbb{R}$ , we write $\Vert \unicode[STIX]{x1D703}\Vert _{\mathbb{R}/\mathbb{Z}}$ for the distance from $\unicode[STIX]{x1D703}$ to the nearest integer, and $e(\unicode[STIX]{x1D703})=\text{e}^{2\unicode[STIX]{x1D70B}\text{i}\unicode[STIX]{x1D703}}$ . Observe from elementary trigonometry that

(2.2) $$\begin{eqnarray}|e(\unicode[STIX]{x1D703})-1|=2|\text{sin}(\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703})|\asymp \Vert \unicode[STIX]{x1D703}\Vert _{\mathbb{R}/\mathbb{Z}}\end{eqnarray}$$

and hence also

(2.3) $$\begin{eqnarray}1-\cos (2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703})=2|\text{sin}(\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703})|^{2}\asymp \Vert \unicode[STIX]{x1D703}\Vert _{\mathbb{ R}/\mathbb{Z}}^{2}.\end{eqnarray}$$

We will also use the triangle inequalities

(2.4) $$\begin{eqnarray}\Vert \unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}\Vert _{\mathbb{R}/\mathbb{Z}}\leqslant \Vert \unicode[STIX]{x1D703}_{1}\Vert _{\mathbb{R}/\mathbb{Z}}+\Vert \unicode[STIX]{x1D703}_{2}\Vert _{\mathbb{R}/\mathbb{Z}};\qquad \Vert k\unicode[STIX]{x1D703}\Vert _{\mathbb{R}/\mathbb{Z}}\leqslant |k|\Vert \unicode[STIX]{x1D703}\Vert _{\mathbb{R}/\mathbb{Z}}\end{eqnarray}$$

for $\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2}\in \mathbb{R}/\mathbb{Z}$ and $k\in \mathbb{Z}$ frequently in the sequel, often without further comment.

For any prime $p$ , we (by slight abuse of notation) let $a\mapsto a/p$ be the obvious homomorphism from $\mathbb{Z}/p\mathbb{Z}$ to $\mathbb{R}/\mathbb{Z}$ that maps $a~(\operatorname{mod}~p)$ to $a/p~(\operatorname{mod}~1)$ for any integer $a$ . We then define $e_{p}:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{C}$ to be the character

$$\begin{eqnarray}e_{p}(a):=e\biggl(\frac{a}{p}\biggr)=\text{e}^{2\unicode[STIX]{x1D70B}\text{i}a/p}\end{eqnarray}$$

of $\mathbb{Z}/p\mathbb{Z}$ .

3 High-level overview of argument

We will establish Theorem 1.1 by establishing the following result, related to the Khinchin-type recurrence theorems mentioned earlier. It will be convenient to introduce the notation

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(\mathbf{f}):=\mathbb{E}\mathbf{f}(\mathbf{a})\mathbf{f}(\mathbf{a}+\mathbf{r})\mathbf{f}(\mathbf{a}+2\mathbf{r})\mathbf{f}(\mathbf{a}+3\mathbf{r})\end{eqnarray}$$

whenever $\mathbf{a},\mathbf{r}$ are random variables on $\mathbb{Z}/p\mathbb{Z}$ and $\mathbf{f}:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ is a random function; of course, the notation can also be applied to deterministic functions $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ . Later on we will also need the conditional variant

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(\mathbf{f}|E):=\mathbb{E}(\mathbf{f}(\mathbf{a})\mathbf{f}(\mathbf{a}+\mathbf{r})\mathbf{f}(\mathbf{a}+2\mathbf{r})\mathbf{f}(\mathbf{a}+3\mathbf{r})|E)\end{eqnarray}$$

for some events $E$ of non-zero probability. Informally, this quantity counts the density of arithmetic progressions $\mathbf{a},\mathbf{a}\,+\,\mathbf{r},\mathbf{a}\,+\,2\mathbf{r},\mathbf{a}\,+\,3\mathbf{r}$ on the event $E$ weighted by $\mathbf{f}$ , where $\mathbf{a},\mathbf{r}$ need not be drawn uniformly or independently (and $\mathbf{f}$ may also be coupled to $\mathbf{a},\mathbf{r}$ ).

Theorem 3.1. Let $p$ be a prime, let $\unicode[STIX]{x1D702}$ be a real number with $0<\unicode[STIX]{x1D702}\leqslant {\textstyle \frac{1}{10}}$ , and let $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ be a function. Then there exist random variables $\mathbf{a},\mathbf{r}\in \mathbb{Z}/p\mathbb{Z}$ , not necessarily independent, obeying the near-uniform distribution bound

(3.2) $$\begin{eqnarray}\mathbb{E}f(\mathbf{a})=\mathbb{E}_{x\in \mathbb{Z}/p\mathbb{Z}}f(x)+O(\unicode[STIX]{x1D702}),\end{eqnarray}$$

the recurrence property

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f)\geqslant (\mathbb{E}f(\mathbf{a}))^{4}-O(\unicode[STIX]{x1D702}),\end{eqnarray}$$

and the “thickness” bound

(3.4) $$\begin{eqnarray}\mathbb{P}(\mathbf{r}=0)\ll \exp (-\unicode[STIX]{x1D702}^{-O(1)})/p.\end{eqnarray}$$

We note that a variant of Theorem 3.1 was established by us in [Reference Green13] (answering a question in [Reference Bergelson, Host and Kra3]), in which the random variable $\mathbf{a}$ was uniformly distributed in $\mathbb{Z}/p\mathbb{Z}$ , the random variable $\mathbf{r}$ was uniformly distributed in a subset of $\mathbb{Z}/p\mathbb{Z}$ of size $\gg _{\unicode[STIX]{x1D702}}p$ and was independent of $\mathbf{a}$ , and the condition (3.4) (which is crucial to the quantitative bound in Theorem 1.1) was not present. Compared to that result, Theorem 3.1 obtains the much more quantitative bound (3.4), but at the expense of no longer enforcing independence between $\mathbf{a}$ and $\mathbf{r}$ . The use of non-independent random variables $\mathbf{a},\mathbf{r}$ is an innovation of this current paper; it is similar to the technique in previous papers of using “factors” (finite partitions) to break up the domain $\mathbb{Z}/p\mathbb{Z}$ into smaller “atoms” such as Bohr sets and analyzing each atom separately. However there will be technical advantages from the more general framework of pairs of independent random variables $\mathbf{a},\mathbf{r}$ . In particular we will be able to avoid some of the boundary issues arising from irregularity of Bohr sets, by using the smoother device of “regular probability distributions” associated to such sets. Although $f$ is allowed to attain negative values in Theorem 3.1, in our applications we shall only be concerned with the case when $f$ is non-negative.

Let us now see how Theorem 1.1 follows from Theorem 3.1. Clearly we may assume that $N\geqslant 100$ . Suppose that $A$ is a subset of $\{1,\ldots ,N\}$ without any non-trivial four-term arithmetic progressions. By Bertrand’s postulate, we may find a prime $p$ between (for example) $2N$ and $4N$ . If we define $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ to be the indicator function $1_{A}$ of $A$ (viewed as a subset of $\mathbb{Z}/p\mathbb{Z}$ ), then we have

(3.5) $$\begin{eqnarray}\mathbb{E}_{x\in \mathbb{Z}/p\mathbb{Z}}f(x)=\frac{|A|}{p}\end{eqnarray}$$

and also

(3.6) $$\begin{eqnarray}f(a)f(a+r)f(a+2r)f(a+3r)=0\end{eqnarray}$$

whenever $a,r\in \mathbb{Z}/p\mathbb{Z}$ with $r$ non-zero. Now let $\mathbf{a},\mathbf{r}$ be as in Theorem 3.1, with $\unicode[STIX]{x1D702}$ to be chosen later. From (3.2), (3.3), (3.5) we have

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f)\geqslant \biggl(\frac{|A|}{p}\biggr)^{4}-O(\unicode[STIX]{x1D702}).\end{eqnarray}$$

But by (3.6), (3.4), the left-hand side is $O(\exp (-\unicode[STIX]{x1D702}^{-O(1)})/p)$ . Setting $\unicode[STIX]{x1D702}:=c\log ^{-c}p$ for a sufficiently small absolute constant $c>0$ , we conclude that

$$\begin{eqnarray}\biggl(\frac{|A|}{p}\biggr)^{4}\ll \log ^{-c}p\end{eqnarray}$$

and hence $A\ll N\log ^{-c/4}N$ , giving Theorem 1.1.

Remark.

As mentioned previously, the arguments in [Reference Green13] established a bound of the form (3.3) with $\mathbf{a}$ and $\mathbf{r}$ independent, and also one could ensure that $\mathbf{a}$ was uniformly distributed over $\mathbb{Z}/p\mathbb{Z}$ . As a consequence, one could establish a variant of Theorem 1.1, namely that for any $N\geqslant 1$ , $\unicode[STIX]{x1D702}>0$ , and $A\subset [N]$ , one had

$$\begin{eqnarray}\frac{|A\cap (A-r)\cap (A-2r)\cap (A-3r)|}{N}\geqslant \biggl(\frac{|A|}{N}\biggr)^{4}-\unicode[STIX]{x1D702}\end{eqnarray}$$

for $\gg _{\unicode[STIX]{x1D702}}N$ choices of $0\leqslant r\leqslant N$ . Unfortunately our methods do not seem to provide a good bound of this form due to our coupling together of $\mathbf{a}$ and $\mathbf{r}$ .

It remains to establish Theorem 3.1. As in [Reference Bergelson, Host and Kra3, Reference Green13], the lower bound (3.3) will ultimately come from the following consequence of the Cauchy–Schwarz inequality that counts solutions to the equation $x-3y+3z-w=0$ for $x,y,z,w$ in some subset of a compact abelian group; this inequality is a specific feature of the theory of length-four progressions that is not available for longer progressionsFootnote ² .

Lemma 3.2 (Application of Cauchy–Schwarz).

Let $G=(G,+)$ be a compact abelian group, let $\unicode[STIX]{x1D707}$ be the probability Haar measure on $G$ , and let $F:G\rightarrow \mathbb{R}$ be a bounded measurable function. Then

$$\begin{eqnarray}\int _{G}\int _{G}\int _{G}F(x)F(y)F(z)F(x-3y+3z)\,d\unicode[STIX]{x1D707}(x)\,d\unicode[STIX]{x1D707}(y)\,d\unicode[STIX]{x1D707}(z)\geqslant \biggl(\int _{G}F\,d\unicode[STIX]{x1D707}\biggr)^{4}.\end{eqnarray}$$

Proof. Making the change of variables $w=x-3y$ and using Fubini’s theorem, the left-hand side may be rewritten as

$$\begin{eqnarray}\int _{G}\biggl(\int _{G}F(w+3y)F(y)\,d\unicode[STIX]{x1D707}(y)\biggr)^{2}\,d\unicode[STIX]{x1D707}(w),\end{eqnarray}$$

which by the Cauchy–Schwarz inequality is at least

$$\begin{eqnarray}\biggl(\int _{G}\int _{G}F(w+3y)F(y)\,d\unicode[STIX]{x1D707}(y)\,d\unicode[STIX]{x1D707}(w)\biggr)^{2}.\end{eqnarray}$$

But by a further application of Fubini’s theorem, the expression inside the square is $(\int _{G}F(x)\,d\unicode[STIX]{x1D707}(x))^{2}$ . The claim follows.◻

To see the relevance of this lemma to Theorem 3.1, and to motivate the strategy of proof of that theorem, let us first test that theorem on some key examples. To simplify the exposition, our discussion will be somewhat non-rigorous in nature; for instance, we will make liberal use of the non-rigorous symbol ${\approx}$ without quantifying the nature of the approximation.

Example 1 (A well-distributed pure quadratic factor).

Let $G$ be the $d$ -torus $G=(\mathbb{R}/\mathbb{Z})^{d}$ for some bounded $d=O(1)$ , and let $F:G\rightarrow [-1,1]$ be a smooth function (independent of $p$ ); for instance, $F$ could be a finite linear combination of characters $\unicode[STIX]{x1D712}:G\rightarrow S^{1}$ of $G$ . Let $\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{d}\in \mathbb{Z}/p\mathbb{Z}$ be “generic” frequencies, in the sense that there are no non-trivial linear relations of the form

(3.7) $$\begin{eqnarray}k_{1}\unicode[STIX]{x1D6FC}_{1}+\cdots +k_{d}\unicode[STIX]{x1D6FC}_{d}=0\end{eqnarray}$$

with $k_{1},\ldots ,k_{d}=O(1)$ not all equal to zero. We also introduce some additional frequencies $\unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{d}\in \mathbb{Z}/p\mathbb{Z}$ , for which we impose no genericity restrictions. Let $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ be the function

$$\begin{eqnarray}f(a):=F(Q(a)),\end{eqnarray}$$

where $Q:\mathbb{Z}/p\mathbb{Z}\rightarrow G$ is the quadratic polynomial

$$\begin{eqnarray}Q(a):=\biggl(\frac{\unicode[STIX]{x1D6FC}_{1}a^{2}+\unicode[STIX]{x1D6FD}_{1}a}{p},\ldots ,\frac{\unicode[STIX]{x1D6FC}_{d}a^{2}+\unicode[STIX]{x1D6FD}_{d}a}{p}\biggr),\end{eqnarray}$$

and where we use the obvious division by zero map $a\mapsto a/p$ from $\mathbb{Z}/p\mathbb{Z}$ to $\mathbb{R}/\mathbb{Z}$ . For any tuples $k=(k_{1},\ldots ,k_{d})\in \mathbb{Z}^{d}\equiv {\hat{G}}$ and $\unicode[STIX]{x1D709}=(\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{d})\in G$ , we define the dot product

$$\begin{eqnarray}k\cdot \unicode[STIX]{x1D709}:=k_{1}\unicode[STIX]{x1D709}_{1}+\cdots +k_{d}\unicode[STIX]{x1D709}_{d}.\end{eqnarray}$$

Because of our genericity hypothesis on the $\unicode[STIX]{x1D6FC}_{i}$ , we see from Gauss sum estimates that

$$\begin{eqnarray}\mathbb{E}_{a\in \mathbb{Z}/p\mathbb{Z}}e(k\cdot Q(a))\approx 0\end{eqnarray}$$

for any bounded tuple $k\in \mathbb{Z}^{d}$ when $p$ is large. By the Weyl equidistribution criterion, we thus see that when $p$ is large, the quantity $(\unicode[STIX]{x1D6FC}a^{2}+\unicode[STIX]{x1D6FD}a)/p$ becomes equidistributed in $G$ as $a$ ranges over $\mathbb{Z}/p\mathbb{Z}$ . In particular, as $F$ was assumed to be smooth, we expect to have

$$\begin{eqnarray}\mathbb{E}f(\mathbf{a})=\mathbb{E}_{a\in \mathbb{Z}/p\mathbb{Z}}f(a)\approx \int _{G}F(x)\,d\unicode[STIX]{x1D707}(x)\end{eqnarray}$$

if $\mathbf{a}$ is drawn uniformly in $\mathbb{Z}/p\mathbb{Z}$ . Now suppose that $\mathbf{r}$ is also drawn uniformly in $\mathbb{Z}/p\mathbb{Z}$ , independently of $\mathbf{a}$ . The tuple

(3.8) $$\begin{eqnarray}(Q(\mathbf{a}),Q(\mathbf{a}+\mathbf{r}),Q(\mathbf{a}+2\mathbf{r}),Q(\mathbf{a}+3\mathbf{r}))\end{eqnarray}$$

will not become equidistributed in $G^{4}$ , because of the elementary algebraic identity

(3.9) $$\begin{eqnarray}Q(\mathbf{a})-3Q(\mathbf{a}+\mathbf{r})+3Q(\mathbf{a}+2\mathbf{r})-Q(\mathbf{a}+3\mathbf{r})=0,\end{eqnarray}$$

which is a discrete version of the fact that the third derivative of any quadratic polynomial vanishes. However, this turns out to be the only constraint on this tuple in the limit $p\rightarrow \infty$ . Indeed, from the genericity hypothesis on the $\unicode[STIX]{x1D6FC}_{i}$ , one can verify that the quadratic form

$$\begin{eqnarray}(a,r)\mapsto k_{0}\cdot Q_{0}(a)+k_{1}\cdot Q_{0}(a+r)+k_{2}\cdot Q_{0}(a+2r)+k_{3}\cdot Q_{0}(a+3r)\end{eqnarray}$$

on $(\mathbb{Z}/p\mathbb{Z})^{2}$ for bounded tuples $k_{0},k_{1},k_{2},k_{3}\in \mathbb{Z}^{d}$ vanishes if and only if $(k_{0},k_{1},k_{2},k_{3})$ is of the form $(k,-3k,3k,-k)$ for some tuple $k$ , where

$$\begin{eqnarray}Q_{0}(a):=\biggl(\frac{\unicode[STIX]{x1D6FC}_{1}a^{2}}{p},\ldots ,\frac{\unicode[STIX]{x1D6FC}_{d}a^{2}}{p}\biggr)\end{eqnarray}$$

denotes the purely quadratic component of $Q(a)$ . Using this and a variant of the Weyl equidistribution criterion, one can eventually compute that

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f)\approx \int _{G}\int _{G}\int _{G}F(x)F(y)F(z)F(x-3y+3z)\,d\unicode[STIX]{x1D707}(x)\,d\unicode[STIX]{x1D707}(y)\,d\unicode[STIX]{x1D707}(z).\end{eqnarray}$$

Applying Lemma 3.2, we conclude (a heuristic version of) Theorem 3.1 in this case, taking $\mathbf{a},\mathbf{r}$ to be independent uniformly distributed variables on $\mathbb{Z}/p\mathbb{Z}$ .

Example 2 (A well-distributed impure quadratic factor).

Now we give a “local” version of the first example, in which the function $f$ exhibits “locally quadratic” behaviour rather than “globally quadratic” behaviour. Let $\unicode[STIX]{x1D702}>0$ be a small parameter, and suppose that $p$ is very large compared to $\unicode[STIX]{x1D702}$ . We suppose that the cyclic group $\mathbb{Z}/p\mathbb{Z}$ is somehow partitioned into a number $P_{1},\ldots ,P_{m}$ of arithmetic progressions; the number $m$ of such progressions should be thought of as being moderately large (e.g. $m\sim \exp (1/\unicode[STIX]{x1D702}^{O(1)})$ for some parameter $\unicode[STIX]{x1D702}>0$ ). Consider one such progression, for example $P_{c}=\{b_{c}+ns_{c}:1\leqslant n\leqslant N_{c}\}$ for some $b_{c},s_{c}\in \mathbb{Z}/p\mathbb{Z}$ and some $N_{c}>0$ ; one should think of $N_{c}$ as being reasonably large, e.g. $N_{c}\gg \exp (-1/\unicode[STIX]{x1D702}^{O(1)})p$ . To each such progression $P_{c}$ , we associate a torus $G_{c}=(\mathbb{R}/\mathbb{Z})^{d_{c}}$ for some bounded $d_{c}$ with probability Haar measure $\unicode[STIX]{x1D707}_{c}$ , a smooth function $F_{c}:G_{c}\rightarrow [-1,1]$ , and a collection $\unicode[STIX]{x1D709}_{c,1},\ldots ,\unicode[STIX]{x1D709}_{c,d_{c}}\in \mathbb{R}/\mathbb{Z}$ of frequencies that are generic in the sense that there does not exist any non-trivial relations of the form

(3.10) $$\begin{eqnarray}k_{1}\unicode[STIX]{x1D709}_{c,1}+\cdots +k_{d_{c}}\unicode[STIX]{x1D709}_{c,d_{c}}=O\biggl(\frac{1}{N_{c}}\biggr)~(\operatorname{mod}~1)\end{eqnarray}$$

for bounded $k_{1},\ldots ,k_{d_{c}}\in \mathbb{Z}$ . We then define the function $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ by setting

$$\begin{eqnarray}f(b_{c}+ns_{c}):=F_{c}(\unicode[STIX]{x1D709}_{c,d_{1}}n^{2},\ldots ,\unicode[STIX]{x1D709}_{c,d_{c}}n^{2})\end{eqnarray}$$

for $1\leqslant c\leqslant m$ and $1\leqslant n\leqslant N_{c}$ . One could also add a lower order linear term to the phases $\unicode[STIX]{x1D709}_{c,i}n^{2}$ , as in the preceding example, if desired, but we will not do so here to simplify the exposition slightly.

Within each progression $P_{c}$ , a Weyl equidistribution analysis (using the genericity hypothesis) reveals that the tuple $(\unicode[STIX]{x1D709}_{c,d_{1}}n^{2},\ldots ,\unicode[STIX]{x1D709}_{c,d_{c}}n^{2})$ becomes equidistributed in $G_{c}$ as $p$ becomes large, so that

(3.11) $$\begin{eqnarray}\mathbb{E}_{a\in P_{c}}f(a)\approx \int _{G_{c}}F_{c}(x)\,d\unicode[STIX]{x1D707}_{c}(x).\end{eqnarray}$$

Now we define the random variables $\mathbf{a},\mathbf{r}\in \mathbb{Z}/p\mathbb{Z}$ as follows. We first select a random element $\mathbf{c}$ from $\{1,\ldots ,m\}$ with $\mathbb{P}(\mathbf{c}=c)=|P_{j}|/p$ for $c=1,\ldots ,m$ . Conditioning on the event that $\mathbf{c}$ is equal to $c$ , we then select $\mathbf{a}$ uniformly at random from $P_{c}$ , and also select $\mathbf{r}$ uniformly at random from an arithmetic progression of the form

(3.12) $$\begin{eqnarray}\{ns_{c}:|n|\leqslant \exp (-1/\unicode[STIX]{x1D702}^{-C})N_{c}\},\end{eqnarray}$$

with $\mathbf{a}$ and $\mathbf{r}$ independent after conditioning on $\mathbf{c}=c$ . Note that $\mathbf{a}$ and $\mathbf{r}$ are only conditionally independent, relative to the auxiliary variable $\mathbf{c}$ ; if one does not perform this conditioning, then $\mathbf{a}$ and $\mathbf{r}$ become coupled to each other through their mutual dependence on $\mathbf{c}$ .

Without conditioning on $\mathbf{c}$ , the random variable $\mathbf{a}$ becomes uniformly distributed on $\mathbb{Z}/p\mathbb{Z}$ , thus

$$\begin{eqnarray}\mathbb{E}f(\mathbf{a})=\mathbb{E}_{a\in \mathbb{Z}/p\mathbb{Z}}f(a).\end{eqnarray}$$

Also, from (3.11) we have the conditional expectation

$$\begin{eqnarray}\mathbb{E}(f(\mathbf{a})|\mathbf{c}=c)\approx \int _{G_{c}}F_{c}(x)\,d\unicode[STIX]{x1D707}_{c}(x).\end{eqnarray}$$

A modification of the equidistribution analysis from the first example also gives

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f|\mathbf{c}=c)\nonumber\\ \displaystyle & & \displaystyle \quad \gtrapprox \int _{G_{c}}\int _{G_{c}}\int _{G_{c}}F_{c}(x)F_{c}(y)F_{c}(z)F(x-3y+3z)\,d\unicode[STIX]{x1D707}_{c}(x)\,d\unicode[STIX]{x1D707}_{c}(y)\,d\unicode[STIX]{x1D707}_{c}(z),\nonumber\end{eqnarray}$$

where the conditional quartic form $\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f|\mathbf{c}=c)$ was defined in (3.1), and hence by Lemma 3.2 we have

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f|\mathbf{c}=c)\gtrapprox (\mathbb{E}(f(\mathbf{a})|\mathbf{c}=c))^{4}.\end{eqnarray}$$

Averaging in $c$ (weighted by $\mathbb{P}(\mathbf{c}=c)$ ) to remove the conditional expectation on the left-hand side, and then applying Hölder’s inequality, we obtain a heuristic version of Theorem 3.1 in this case.

Example 3 (A poorly distributed pure quadratic factor).

We now return to the situation of the first example, except that we no longer impose the genericity hypothesis, that is to say we allow for a non-trivial relation of the form (3.7). Without loss of generality we can take the coefficient $k_{d}$ of this relation to be non-zero. Because of this relation, the quantity $Q(\mathbf{a})$ studied in the first example and the tuple (3.8) may not necessarily be as equidistributed as before. However, we can use this irregularity of distribution to modify the representation of $f$ (up to a small error) in such a manner as to reduce the number $d$ of quadratic phases involved. Namely, we can write

$$\begin{eqnarray}f(a):=\tilde{F}\biggl(\tilde{Q}(a),\frac{\unicode[STIX]{x1D6FE}a}{p}\biggr)\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle \tilde{Q}(a) & := & \displaystyle \biggl(\frac{k_{d}^{-1}\unicode[STIX]{x1D6FC}_{1}a^{2}+k_{d}^{-1}\unicode[STIX]{x1D6FD}_{1}a}{p},\ldots ,\frac{k_{d}^{-1}\unicode[STIX]{x1D6FC}_{d-1}a^{2}+k_{d}^{-1}\unicode[STIX]{x1D6FD}_{d-1}a}{p}\biggr),\nonumber\\ \displaystyle \unicode[STIX]{x1D6FE} & := & \displaystyle \unicode[STIX]{x1D6FD}_{d}+k_{1}k_{d}^{-1}\unicode[STIX]{x1D6FD}_{1}+\cdots +k_{d-1}k_{d}^{-1}\unicode[STIX]{x1D6FD}_{d-1},\nonumber\\ \displaystyle \tilde{F}(x_{1},\ldots ,x_{d-1},y) & := & \displaystyle F(k_{d}x_{1},\ldots ,k_{d}x_{d-1},-k_{1}x_{1}-\cdots -k_{d-1}x_{d-1}+y)\nonumber\end{eqnarray}$$

and where we take advantage of the field structure of $\mathbb{Z}/p\mathbb{Z}$ to locate an inverse $k_{d}^{-1}$ of $k_{d}$ in this field. For our quantitative analysis we will run into a technical difficulty with this representation, in that the Lipschitz constant of $\tilde{F}$ will increase by an undesirable amount compared to that of $F$ when one performs this change of variable, at least if one uses the standard metric on the torus. To fix this, we will eventually have to work with more general tori $\prod _{i=1}^{d}\mathbb{R}/\unicode[STIX]{x1D706}_{i}\mathbb{Z}$ than the standard torus $(\mathbb{R}/\mathbb{Z})^{d}$ , but we ignore this issue for now to continue with the heuristic discussion.

To remove the dependence on the linear phase $\unicode[STIX]{x1D6FE}a/p$ , we partition $\mathbb{Z}/p\mathbb{Z}$ into “(shifted) Bohr sets” $B_{1},\ldots ,B_{m}$ for some moderately large $m$ (e.g. $m\sim \exp (1/\unicode[STIX]{x1D702}^{-C})$ for some constant $C>0$ ), defined by

$$\begin{eqnarray}B_{c}:=\biggl\{a\in \mathbb{Z}/p\mathbb{Z}:\frac{\unicode[STIX]{x1D6FE}a}{p}\in \biggl[\frac{c-1}{m},\frac{c}{m}\biggr)~(\operatorname{mod}~1)\biggr\}\end{eqnarray}$$

for $c=1,\ldots ,m$ . On each Bohr set $B_{c}$ , we have the approximation

$$\begin{eqnarray}f(a):=\tilde{F}_{c}(\tilde{Q}(a))\end{eqnarray}$$

where $\tilde{F}_{c}(x,y):=\tilde{F}(x,c/m)$ . Using the heuristic that Bohr sets behave like arithmetic progressions, the situation is now similar to that in the second example, with the number of quadratic phases involved reduced from $d$ to $d-1$ , except that there may still be some non-trivial relations among the surviving quadratic phases (and one also now has some lower order linear terms in the quadratic phases). To deal with this difficulty, we turn now to the consideration of yet another example.

Example 5 (A pseudorandom perturbation of a pure quadratic factor).

In all the preceding examples, the function $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ under consideration was “locally quadratically structured”, in the sense that on local regions such as $P_{c}$ , the function $f$ could be accurately represented in terms of quadratic phase functions $a\mapsto Q(a)$ . This is however not the typical behaviour expected for a general function $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ . A more representative example would be a function of the form

$$\begin{eqnarray}f(a):=f_{1}(a)+f_{2}(a),\end{eqnarray}$$

where $f_{1}:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{R}$ is a function of the type considered in the first example, thus

$$\begin{eqnarray}f_{1}(a)=F(Q(a))\end{eqnarray}$$

for some quadratic function $Q:\mathbb{Z}/p\mathbb{Z}\rightarrow G$ into a torus $G=(\mathbb{R}/\mathbb{Z})^{d}$ and some smooth $F:G\rightarrow [-1,1]$ , and $f_{2}:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ is a function that is globally Gowers uniform in the sense that

(3.13) $$\begin{eqnarray}\mathbb{E}\mathop{\prod }_{(\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3})\in \{0,1\}^{3}}f_{2}(\mathbf{a}+\unicode[STIX]{x1D714}_{1}\mathbf{h}_{1}+\unicode[STIX]{x1D714}_{2}\mathbf{h}_{2}+\unicode[STIX]{x1D714}_{3}\mathbf{h}_{3})\approx 0,\end{eqnarray}$$

where $\mathbf{a},\mathbf{h}_{1},\mathbf{h}_{2},\mathbf{h}_{3}$ are drawn independently and uniformly at random from $\mathbb{Z}/p\mathbb{Z}$ . A typical example to keep in mind is when $F$ (and hence $f_{1}$ ) takes values in $[0,1]$ , and $f=\mathbf{f}$ is a random function with $f(a)$ equal to $1$ with probability $f_{1}(a)$ and $0$ with probability $1-f_{1}(a)$ , independently as $a\in \mathbb{Z}/p\mathbb{Z}$ varies; then the $f_{2}(a)$ for $a\in \mathbb{Z}/p\mathbb{Z}$ become independent random variables of mean zero, and the global Gowers uniformity can be established with high probability using tools such as the Chernoff inequality.

From the standard theory of the Gowers norms (see e.g. [Reference Tao and Vu33, Ch. 11]), one can use the global Gowers uniformity of $f_{2}$ , combined with a number of applications of the Cauchy–Schwarz inequality, to establish a “generalized von Neumann theorem” that, in our current context, implies that $f$ and $f_{1}$ globally count approximately the same number of length-four progressions in the sense that

(3.14) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f)\approx \unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f_{1});\end{eqnarray}$$

similarly one also has

(3.15) $$\begin{eqnarray}\mathbb{E}f(\mathbf{a})\approx \mathbb{E}f_{1}(\mathbf{a}).\end{eqnarray}$$

As a consequence, Theorem 3.1 for such functions follows (heuristically, at least) from the analysis of the first example, at least if one assumes the genericity of the frequencies $\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{d}$ .

Example 6 (A pseudorandom perturbation of an impure quadratic factor).

We now consider a situation that is to the second example as the fifth example was to the first. Namely, we consider a function of the form

$$\begin{eqnarray}f(a):=f_{1}(a)+f_{2}(a),\end{eqnarray}$$

where $f_{1}:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ is a function of the type considered in the second example, thus

$$\begin{eqnarray}f_{1}(b_{c}+ns_{c}):=F_{c}(\unicode[STIX]{x1D709}_{c,d_{1}}n^{2},\ldots ,\unicode[STIX]{x1D709}_{c,d_{c}}n^{2})\end{eqnarray}$$

for $c=1,\ldots ,m$ and $n=1,\ldots ,N_{c}$ . As for the function $f_{2}:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ , global Gowers uniformity of $f_{2}$ will be too weak of a hypothesis for our purposes, because the random variable $\mathbf{r}$ appearing in the second example is now localized to a significantly smaller region than $\mathbb{Z}/p\mathbb{Z}$ . Instead, we will require the local Gowers uniformity hypothesis

(3.16) $$\begin{eqnarray}\mathbb{E}\mathop{\prod }_{(\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3})\in \{0,1\}^{3}}f_{2}(\mathbf{a}+\unicode[STIX]{x1D714}_{1}\mathbf{h}_{1}+\unicode[STIX]{x1D714}_{2}\mathbf{h}_{2}+\unicode[STIX]{x1D714}_{3}\mathbf{h}_{3})\approx 0,\end{eqnarray}$$

where $\mathbf{a}$ is now the random variable from the second example (in particular, $\mathbf{a}$ depends on the auxiliary random variable $\mathbf{c}$ ), and once one conditions on an event $\mathbf{c}=c$ for $c=1,\ldots ,m$ , one draws $\mathbf{h}_{1},\mathbf{h}_{2},\mathbf{h}_{3}$ independently of each other and from $\mathbf{a}$ , and each $\mathbf{h}_{i}$ drawn uniformly from an arithmetic progression of the form

(3.17) $$\begin{eqnarray}\{ns_{c}:|n|\leqslant \exp (-1/\unicode[STIX]{x1D702}^{-C_{i}})N_{c}\},\end{eqnarray}$$

for some constant $C_{i}>0$ (for technical reasons, it is convenient to allow these constants $C_{1},C_{2},C_{3}$ to be different from each other, and also to be larger than the constant $C$ appearing in (3.12), so that $\mathbf{h}_{1},\mathbf{h}_{2},\mathbf{h}_{3}$ range over a narrower scale than $\mathbf{r}$ ). As with $\mathbf{a}$ and $\mathbf{r}$ , the random variables $\mathbf{a},\mathbf{h}_{1},\mathbf{h}_{2},\mathbf{h}_{3}$ are now only conditionally independent relative to the auxiliary variable $\mathbf{c}$ , but are not independent of each other without this conditioning, as they are coupled to each other through $\mathbf{c}$ .

As it turns out, once one assumes this local Gowers uniformity of $f_{2}$ , one can modify the Cauchy–Schwarz arguments used to establish the global generalized von Neumann theorem to obtain the approximations (3.14), (3.15) for the random variables $\mathbf{a},\mathbf{r}$ considered in the second example, at which point Theorem 3.1 for this choice of $f$ follows (heuristically, at least) from the analysis of that example, at least if one assumes that there are no non-trivial relations of the form (3.10).

Example 7 (Non-pseudorandom perturbation of a pure quadratic factor).

We now modify the fifth example by replacing the hypothesis (3.13) by its negation

(3.18) $$\begin{eqnarray}\mathbb{E}\mathop{\prod }_{(\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3})\in \{0,1\}^{3}}f_{2}(\mathbf{a}+\unicode[STIX]{x1D714}_{1}\mathbf{h}_{1}+\unicode[STIX]{x1D714}_{2}\mathbf{h}_{2}+\unicode[STIX]{x1D714}_{3}\mathbf{h}_{3})\gg 1\end{eqnarray}$$

(it is not difficult to show that the left-hand side is non-negative). In this case, the generalized von Neumann theorem used in that example does not give a good estimate. However, in this situation one can apply the inverse theorem for the Gowers norm established by us in [Reference Green and Tao14]. To obtain good quantitative bounds, we will use the version of that theorem that involves local correlation with quadratic objects (as opposed to a somewhat weak global correlation with a single “locally quadratic” object). Namely, if (3.18) holds, then one can partition $\mathbb{Z}/p\mathbb{Z}$ into a moderately large (e.g. $O(\exp (1/\unicode[STIX]{x1D702}^{-O(1)}))$ ) number of pieces $P_{1},\ldots ,P_{m}$ , such that on each piece $P_{c}$ , the function $f_{2}$ correlates with a “quadratically structured” object. The precise statement is somewhat technical to state, but one simple special case of this conclusion is that the pieces $P_{1},\ldots ,P_{m}$ are arithmetic progressions as in the second example, and for a “significant number” of the progressions

$$\begin{eqnarray}P_{c}=\{b_{c}+ns_{c}:1\leqslant n\leqslant N_{c}\}\end{eqnarray}$$

there exists a frequency $\unicode[STIX]{x1D709}_{c}\in \mathbb{R}/\mathbb{Z}$ such that

$$\begin{eqnarray}|\mathbb{E}_{1\leqslant n\leqslant N_{c}}f_{2}(b_{c}+ns_{c})e(-\unicode[STIX]{x1D709}_{c}n^{2})|\gg 1.\end{eqnarray}$$

(In general, one would take $P_{c}$ to be Bohr sets of moderately high rank, rather than arithmetic progressions, and the phase $a\mapsto \unicode[STIX]{x1D709}_{c}a^{2}/p$ would have to be replaced by a more general locally quadratic phase function on such a Bohr set, but we ignore these technicalities for the current informal discussion.) From this and the cosine rule, it is possible to find a function $g:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ that is equal to (the real part of) a scalar multiple of the quadratic phases $b_{c}+ns_{c}\mapsto e(\unicode[STIX]{x1D709}_{c}n^{2})$ on each progression $P_{c}$ , such that $f_{2}+g$ has an energy decrement compared to $f_{2}$ in the sense that

(3.19) $$\begin{eqnarray}\mathbb{E}_{a\in \mathbb{Z}/p\mathbb{Z}}(f_{2}(a)+g(a))^{2}\leqslant \mathbb{E}_{a\in \mathbb{Z}/p\mathbb{Z}}f_{2}(a)^{2}-\unicode[STIX]{x1D702}^{C}\end{eqnarray}$$

for some constant $C>0$ . In this situation, we can modify the decomposition $f=f_{1}+f_{2}$ by adding $g$ to $f_{2}$ and subtracting it from $f_{1}$ . (Strictly speaking, this may make $f_{1}$ and $f_{2}$ range slightly outside of $[-1,1]$ , but because $f$ itself ranges in $[-1,1]$ , it turns out to be relatively easy to modify $f_{1},f_{2}$ further to rectify this problem.) The new function $f_{1}$ has a similar “quadratic structure” to the previous function $f_{1}$ , except that the quadratic structure is now localized to the cells $P_{1},\ldots ,P_{m}$ of the partition of $\mathbb{Z}/p\mathbb{Z}$ , and the number of quadratic functions has been increased by one. If the new function $f_{2}$ is now locally Gowers uniform in the sense of (3.16), then we are now essentially in the situation of the sixth example (at least if there are no non-trivial relations of the form (3.10)), and we can (heuristically at least) conclude Theorem 3.1 in this case by the previous analysis. If $f_{2}$ is locally Gowers uniform but there are additionally some relations of the form (3.10), then one can hope to adapt the analysis of the fourth example to reduce the quadratic complexity of $f_{1}$ on all the poorly distributed cells, at which point one restarts the analysis. If however $f_{2}$ remains non-uniform, then we need to argue using the analysis of the next and final example.

Example 8 (Non-pseudorandom perturbation of an impure quadratic factor).

Our final and most difficult example will be as to the sixth example as the seventh example was to the fifth. Namely, we modify the sixth example by assuming that the negation of (3.16) holds. Equivalently, one has the lower bound

(3.20) $$\begin{eqnarray}\mathbb{E}\biggl(\mathop{\prod }_{(\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3})\in \{0,1\}^{3}}f_{2}(\mathbf{a}+\unicode[STIX]{x1D714}_{1}\mathbf{h}_{1}+\unicode[STIX]{x1D714}_{2}\mathbf{h}_{2}+\unicode[STIX]{x1D714}_{3}\mathbf{h}_{3})|\mathbf{c}=c\biggr)\gg 1\end{eqnarray}$$

on the local Gowers norm for a “significant fraction” of the $c=1,\ldots ,m$ .

At the qualitative level, the inverse theorem in [Reference Green and Tao14] for the global Gowers norm allows one to also deduce a similar conclusion starting from the hypothesis (3.20). However, the quantitative bounds obtained by this approach turn out to be too poor for the purposes of establishing Theorems 3.1 or 1.1. Instead, one must obtain a quantitative local inverse theorem for the Gowers norm that has reasonably good bounds (of polynomial type) on the amount of correlation that is (locally) attained. Establishing such a theorem is by far the most complicated and lengthy component of this paper, although broadly speaking it follows the same strategy as previous theorems of this type in [Reference Gowers11, Reference Green and Tao14]. If one takes this local inverse theorem for granted, then roughly speaking what we can then conclude from the hypothesis (3.20) is that for a significant number of $c=1,\ldots ,m$ , one can partition the cell $P_{c}$ into subcells $P_{c,1},\ldots ,P_{c,m_{c}}$ , and locate a “locally quadratic phase function” $\unicode[STIX]{x1D719}_{c,i}:P_{c,i}\rightarrow \mathbb{R}/\mathbb{Z}$ on each such subcell (generalizing the functions $b_{c}+ns_{c}\mapsto e(\unicode[STIX]{x1D709}_{c}n^{2})$ from the previous example), such that

$$\begin{eqnarray}|\mathbb{E}_{a\in P_{c,i}}f_{2}(b_{c,i})e(-\unicode[STIX]{x1D719}_{c,i}(a))|\gg 1\end{eqnarray}$$

for a significant fraction of the $c,i$ . Using this, one can again obtain an energy decrement of the form (3.19), where now $g$ is (the real part of) a scalar multiple of the functions $a\mapsto e(\unicode[STIX]{x1D719}_{c,i}(a))$ on each $P_{c,i}$ . By arguing as in the sixth example, one can then modify $f_{1}$ and $f_{2}$ in such a way that the “energy” $\mathbb{E}f_{2}(\mathbf{a})^{2}$ decreases significantly, while $f_{1}$ is now locally quadratically structured on a somewhat finer partition of $\mathbb{Z}/p\mathbb{Z}$ than the original partition $P_{1},\ldots ,P_{m}$ , with the number of quadratic phases needed to describe $f_{1}$ on each partition having increased by one. If the function $f_{2}$ is now locally Gowers uniform (with respect to a new set of random variables $\mathbf{a},\mathbf{r}$ adapted to this finer partition), and there are no non-trivial relations of the form we can now (heuristically) conclude Theorem 3.1 from the analysis of the sixth example, assuming the addition of the new quadratic phase has not introduced relations of the form (3.10). If such relations occur, though, one can hope to adapt the analysis of the fourth example to reduce the quadratic complexity of the poorly distributed cells, perhaps at the cost of further subdivision of the cells. Finally, if the new version of $f_{2}$ remains non-uniform with respect to the finer partition, then one iterates the analysis of this example to reduce the energy of $f_{2}$ further. This process cannot continue indefinitely due to the non-negativity of the energy (and also because none of the other steps in the iteration will cause a significant increase in energy). Because of this, one can hope to cover all cases of Theorem 3.1 by some complicated iteration of the eight arguments described above.

Having informally discussed the eight key examples for Theorem 3.1, we return now to the task of proving this theorem rigorously.

It will be convenient to work throughout the rest of the paper with a fixed choice

$$\begin{eqnarray}1<C_{1}<C_{2}<\cdots <C_{5}\end{eqnarray}$$

of absolute constants, with each $C_{i}$ assumed to be sufficiently large depending on the previous $C_{1},\ldots ,C_{i-1}$ . For instance, for sake of concreteness one could choose $C_{i}:=2^{2^{100i}}$ ; of course, other choices are possible. The implied constants in the $O(\,)$ notation will not depend on the $C_{i}$ unless otherwise specified. These constants will serve as exponents for various scales $\unicode[STIX]{x1D702}^{-C_{i}}$ that will appear in our analysis, with the point being that any scale of the form $\unicode[STIX]{x1D702}^{-C_{i}}$ for $i=2,\ldots ,5$ is extremely tiny with respect to any polynomial combination of the previous scales $\unicode[STIX]{x1D702}^{-C_{1}},\ldots ,\unicode[STIX]{x1D702}^{-C_{i-1}}$ .

In all of the eight examples considered above, the function $f$ was approximated by some “quadratically structured” function, usually denoted $f_{1}$ , with the approximation being accurate in various senses with respect to some pair $(\mathbf{a},\mathbf{r})$ of random variables. The rigorous argument will similarly approximate $f$ by a quadratically structured object; it will be convenient to make this object a random function $\mathbf{f}$ rather than a deterministic one (though as it turns out, this function will become deterministic again once an auxiliary random variable $\mathbf{c}$ is fixed). The precise definition of “quadratically structured” will be rather technical, and will eventually be given in Definition 6.1. For now, we shall abstract the properties of “quadratic structure” that we will need, in the following proposition involving an abstract directed graph $G=(V,E)$ (encoding the “structured local approximants”), which we will construct more explicitly later. We will shortly iterate this proposition to establish Theorem 3.1 and hence Theorem 1.1.

Proposition 3.3 (Main proposition, abstract form).

Let $\unicode[STIX]{x1D702}$ be a real number with $0<\unicode[STIX]{x1D702}\leqslant {\textstyle \frac{1}{10}}$ , and let $p$ be a prime with

(3.21) $$\begin{eqnarray}p\geqslant \exp (\unicode[STIX]{x1D702}^{-3C_{5}}).\end{eqnarray}$$

Let $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [0,1]$ be a function. Then there exist the following:

1. (a) a (possibly infinite) directed graph $G=(V,E)$ , with elements $v\in V$ referred to as structured local approximants, and the notation $v\rightarrow v^{\prime }$ used to denote the existence of a directed edge from one structured local approximant $v$ to another $v^{\prime }$ ;

2. (b) a triple $(\mathbf{a}_{v},\mathbf{r}_{v},\mathbf{f}_{v})$ associated to $f$ and to each structured local approximant $v\in V$ , where $\mathbf{a}_{v},\mathbf{r}_{v}$ are random variables in $\mathbb{Z}/p\mathbb{Z}$ , and $\mathbf{f}_{v}:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ is a random function (with $\mathbf{a}_{v},\mathbf{r}_{v},\mathbf{f}_{v}$ not assumed to be independent);

3. (c) a quadratic dimension $d_{2}(v)\in \mathbb{N}$ assigned to each vertex $v\in V$ ;

4. (d) a poorly distributed quadratic dimension $d_{2}^{\text{poor}}(v)\in \mathbb{N}$ assigned to each vertex $v\in V$ , with $0\leqslant d_{2}^{\text{poor}}(v)\leqslant d_{2}(v)$ ; and

5. (e) an initial approximant $v_{0}\in V$ , with $d_{2}(v_{0})=0$ (and hence $d_{2}^{\text{poor}}(v_{0})=0$ ).

Furthermore, whenever a structured local approximant $v_{k}\in V$ can be reached from $v_{0}$ by a path $v_{0}\rightarrow v_{1}\rightarrow \cdots \rightarrow v_{k}$ with $0\leqslant k\leqslant 8\unicode[STIX]{x1D702}^{-2C_{2}}$ , then the following properties are obeyed:

1. (i) one has the “thickness” condition

   (3.22) $$\begin{eqnarray}\mathbb{P}(\mathbf{r}_{v_{k}}=0)\ll \exp (3\unicode[STIX]{x1D702}^{-C_{5}})/p;\end{eqnarray}$$

2. (ii) we have the almost uniformity condition

   (3.23) $$\begin{eqnarray}|\mathbb{E}f(\mathbf{a}_{v_{k}})-\mathbb{E}_{a\in \mathbb{Z}/p\mathbb{Z}}f(a)|\leqslant \unicode[STIX]{x1D702};\end{eqnarray}$$

3. (iii) bad approximation implies energy decrement: if

   (3.24) $$\begin{eqnarray}|\mathbb{E}\mathbf{f}_{v_{k}}(\mathbf{a}_{v_{k}})-f(\mathbf{a}_{v_{k}})|>\unicode[STIX]{x1D702}\end{eqnarray}$$
   or
   (3.25) $$\begin{eqnarray}|\unicode[STIX]{x1D6EC}_{\mathbf{a}_{v_{k}},\mathbf{r}_{v_{k}}}(\mathbf{f}_{v_{k}})-\unicode[STIX]{x1D6EC}_{\mathbf{a}_{v_{k}},\mathbf{r}_{v_{k}}}(f)|>\unicode[STIX]{x1D702}\end{eqnarray}$$
   then there exists a structured local approximant $v_{k+1}\in V$ with $v_{k}\rightarrow v_{k+1}$ such that
   $$\begin{eqnarray}\mathbb{E}|f(\mathbf{a}_{v_{k+1}})-\mathbf{f}_{v_{k+1}}(\mathbf{a}_{v_{k+1}})|^{2}\leqslant \mathbb{E}|f(\mathbf{a}_{v_{k}})-\mathbf{f}_{v_{k}}k(\mathbf{a}_{v_{k}})|^{2}-\unicode[STIX]{x1D702}^{C_{2}}\end{eqnarray}$$
   and
   $$\begin{eqnarray}d_{2}(v_{k+1})\leqslant d_{2}(v_{k})+1.\end{eqnarray}$$

4. (iv) failure of “Khinchin-type recurrence” implies dimension decrement: if

   (3.26) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a}_{v_{k}},\mathbf{r}_{v_{k}}}(\mathbf{f}_{v_{k}})\leqslant (\mathbb{E}\mathbf{f}_{v_{k}}(\mathbf{a}_{v_{k}}))^{4}-\unicode[STIX]{x1D702},\end{eqnarray}$$
   then there exists a structured local approximant $v_{k+1}\in V$ with $v_{k}\rightarrow v_{k+1}$ obeying the bounds
   $$\begin{eqnarray}\displaystyle \mathbb{E}|f(\mathbf{a}_{v_{k+1}})-\mathbf{f}_{v_{k+1}}(\mathbf{a}_{v_{k+1}})|^{2} & {\leqslant} & \displaystyle \mathbb{E}|f(\mathbf{a}_{v_{k}})-\mathbf{f}_{v_{k}}(\mathbf{a}_{v_{k}})|^{2}+\unicode[STIX]{x1D702}^{3C_{2}},\nonumber\\ \displaystyle d_{2}(v_{k+1}) & {\leqslant} & \displaystyle d_{2}(v_{k}),\nonumber\\ \displaystyle d_{2}^{\text{poor}}(v_{k+1}) & {\leqslant} & \displaystyle d_{2}^{\text{poor}}(v_{k})-1.\nonumber\end{eqnarray}$$

The proof of this proposition will occupy the remainder of the paper. For now, let us see how this proposition implies Theorem 3.1. Let $p,\unicode[STIX]{x1D702},f$ be as in that theorem, and let $C_{1},\ldots ,C_{5}$ be as above. If the largeness criterion (3.21) fails, then we may set $\mathbf{r}:=0$ , $\mathbf{f}:=f$ , and draw $\mathbf{a}$ uniformly at random from $\mathbb{Z}/p\mathbb{Z}$ , and it is easy to see that the conclusions of Theorem 3.1 are obeyed (with (3.3) following from Hölder’s inequality). Thus we may assume without loss of generality that (3.21) holds.

Let $G=(V,E)$ , $v_{0}$ , $d_{2}(\,)$ , $d_{2}^{\text{poor}}(\,)$ , and $(\mathbf{a}_{v},\mathbf{r}_{v},\mathbf{f}_{v})$ be as in Proposition 3.3. Suppose first that there exists a structured local approximant $v_{k}\in V$ that can be reached from $v_{0}$ by a path of length at most $8\unicode[STIX]{x1D702}^{-2C_{2}}$ , and for which none of the inequalities (3.24)–(3.26) hold, that is to say one has the bounds

(3.27) $$\begin{eqnarray}\displaystyle & \displaystyle |\mathbb{E}\mathbf{f}_{v_{k}}(\mathbf{a}_{v_{k}})-f_{v_{k}}(\mathbf{a}_{v_{k}})|\leqslant \unicode[STIX]{x1D702}, & \displaystyle\end{eqnarray}$$

(3.28) $$\begin{eqnarray}\displaystyle & \displaystyle |\unicode[STIX]{x1D6EC}_{\mathbf{a}_{v_{k}},\mathbf{r}_{v_{k}}}(\mathbf{f}_{v_{k}})-\unicode[STIX]{x1D6EC}_{\mathbf{a}_{v_{k}},\mathbf{r}_{v_{k}}}(f_{v_{k}})|\leqslant \unicode[STIX]{x1D702} & \displaystyle\end{eqnarray}$$

(3.29) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6EC}_{\mathbf{a}_{v_{k}},\mathbf{r}_{v_{k}}}(\mathbf{f}_{v_{k}})>(\mathbb{E}\mathbf{f}_{v_{k}}(\mathbf{a}_{v_{k}}))^{4}-\unicode[STIX]{x1D702}. & \displaystyle\end{eqnarray}$$

From (3.29), (3.28), (3.27) and the triangle inequality (and the boundedness of $\mathbf{f}_{v_{k}},f$ ) we conclude that

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a}_{v_{k}},\mathbf{r}_{v_{k}}}(f_{v_{k}})>(\mathbb{E}f(\mathbf{a}_{v_{k}}))^{4}-O(\unicode[STIX]{x1D702});\end{eqnarray}$$

combining this with (3.22) and (3.23) we see that the random variables $\mathbf{a}_{v_{k}},\mathbf{r}_{v_{k}}$ obey the properties required of Theorem 3.1. Thus we may assume for sake of contradiction that this situation never occurs, which by Proposition 3.3 implies that whenever $v_{k}\in V$ is a structured local approximant that can be reached from $v_{0}$ by a path of length at most $8\unicode[STIX]{x1D702}^{-2C_{2}}$ , then the conclusions of at least one of (iii) and (iv) hold. Iterating this we may therefore construct a path

$$\begin{eqnarray}v_{0}\rightarrow v_{1}\rightarrow \cdots \rightarrow v_{k_{0}+1}\end{eqnarray}$$

with

(3.30) $$\begin{eqnarray}k_{0}:=\lfloor 8\unicode[STIX]{x1D702}^{-2C_{2}}\rfloor ,\end{eqnarray}$$

such that for every $0\leqslant k\leqslant k_{0}$ , one either has the energy decrement bounds

$$\begin{eqnarray}\displaystyle \mathbb{E}|f(\mathbf{a}_{v_{k+1}})-\mathbf{f}_{k+1}(\mathbf{a}_{v_{k+1}})|^{2} & {\leqslant} & \displaystyle \mathbb{E}|f(\mathbf{a}_{v_{k}})-\mathbf{f}_{k}(\mathbf{a}_{v_{k}})|^{2}-\unicode[STIX]{x1D702}^{C_{2}},\nonumber\\ \displaystyle d_{2}(v_{k+1}) & {\leqslant} & \displaystyle d_{2}(v_{k})+1\nonumber\end{eqnarray}$$

or the dimension decrement bounds

$$\begin{eqnarray}\displaystyle \mathbb{E}|f(\mathbf{a}_{v_{k+1}})-\mathbf{f}_{k+1}(\mathbf{a}_{v_{k+1}})|^{2} & {\leqslant} & \displaystyle \mathbb{E}|f(\mathbf{a}_{v_{k}})-\mathbf{f}_{k}(\mathbf{a}_{v_{k}})|^{2}+\unicode[STIX]{x1D702}^{3C_{2}},\nonumber\\ \displaystyle d_{2}(v_{k+1}) & {\leqslant} & \displaystyle d_{2}(v_{k}),\nonumber\\ \displaystyle d_{2}^{\text{poor}}(v_{k+1}) & {\leqslant} & \displaystyle d_{2}^{\text{poor}}(v_{k})-1.\nonumber\end{eqnarray}$$

Since $v_{0}$ already has the minimum quadratic dimension $d_{2}^{\text{poor}}(v_{0})=0$ , we see that we must experience an energy decrement at the $k=0$ stage. Also, if $k$ is the $j$ th index to experience an energy decrement, we see that $d_{2}^{\text{poor}}(v_{k+1})\leqslant d_{2}(v_{k+1})\leqslant j$ , and so one can have at most $j$ consecutive dimension decrements after the $k$ th stage; in other words, we must experience another energy decrement within $j+1$ steps. By definition of $k_{0}$ , we have $\sum _{0\leqslant j\leqslant 2\unicode[STIX]{x1D702}^{-C_{2}}}(j+1)<k_{0}$ if $C_{2}$ is large enough. We conclude that at least $2\unicode[STIX]{x1D702}^{-C_{2}}$ energy decrements occur within the path $v_{0}\rightarrow \cdots \rightarrow v_{k_{0}+1}$ . This implies that

$$\begin{eqnarray}\mathbb{E}|f(\mathbf{a}_{v_{k_{0}+1}})-\mathbf{f}_{k_{0}+1}(\mathbf{a}_{v_{k_{0}+1}})|^{2}\leqslant \mathbb{E}|f(\mathbf{a}_{v_{0}})-\mathbf{f}_{k+1}(\mathbf{a}_{v_{0}})|^{2}-(2\unicode[STIX]{x1D702}^{-C_{2}})\unicode[STIX]{x1D702}^{C_{2}}+k_{0}\unicode[STIX]{x1D702}^{3C_{2}}.\end{eqnarray}$$

But if $C_{2}$ is sufficiently large, this implies from (3.30) that

$$\begin{eqnarray}\mathbb{E}|f(\mathbf{a}_{v_{k_{0}+1}})-\mathbf{f}_{k_{0}+1}(\mathbf{a}_{v_{k_{0}+1}})|^{2}<\mathbb{E}|f(\mathbf{a}_{v_{0}})-\mathbf{f}_{0}(\mathbf{a}_{v_{0}})|^{2}-4\end{eqnarray}$$

(for example), which leads to a contradiction because the left-hand side is clearly non-negative, and the right-hand side non-positive. This gives the desired contradiction that establishes Theorem 3.1 and hence Theorem 1.1.

It remains to establish Proposition 3.3. This will occupy the remaining portions of the paper.

4 Bohr sets

To define and manipulate the “structured local approximants” that appear in Proposition 3.3, we will need to develop the theory of two mathematical objects. The first is that of a Bohr set, which will be covered in this section; the second is that of a dilated torus, which we will discuss in the next section.

Definition 4.1 (Bohr set).

A subset $S$ of $\mathbb{Z}/p\mathbb{Z}$ is said to be non-degenerate if it contains at least one non-zero element. In this case we define the dual $S$ -norm

$$\begin{eqnarray}\Vert a\Vert _{S^{\bot }}:=\sup _{\unicode[STIX]{x1D709}\in S}\biggl\|\frac{a\unicode[STIX]{x1D709}}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}\end{eqnarray}$$

for any $a\in \mathbb{Z}/p\mathbb{Z}$ , and then define the Bohr set $B(S,\unicode[STIX]{x1D70C})\subset \mathbb{Z}/p\mathbb{Z}$ for any $\unicode[STIX]{x1D70C}>0$ by the formula

$$\begin{eqnarray}B(S,\unicode[STIX]{x1D70C}):=\{a\in \mathbb{Z}/p\mathbb{Z}:\Vert a\Vert _{S^{\bot }}<\unicode[STIX]{x1D70C}\}\end{eqnarray}$$

where $\Vert \unicode[STIX]{x1D703}\Vert _{\mathbb{R}/\mathbb{Z}}$ denotes the distance from $\unicode[STIX]{x1D703}$ to the nearest integer. We refer to $S$ as the set of frequencies of the Bohr set, $\unicode[STIX]{x1D70C}$ as the radius, and $|S|$ as the rank of the Bohr set. We also define the shifted Bohr sets

$$\begin{eqnarray}n+B(S,\unicode[STIX]{x1D70C}):=\{a+n:a\in B(S,\unicode[STIX]{x1D70C})\}\end{eqnarray}$$

for any $n\in \mathbb{Z}/p\mathbb{Z}$ .

From (2.4) we have the triangle inequalities

(4.1) $$\begin{eqnarray}\Vert a+b\Vert _{S^{\bot }}\leqslant \Vert a\Vert _{S^{\bot }}+\Vert b\Vert _{S^{\bot }};\qquad \Vert ka\Vert _{S^{\bot }}\leqslant |k|\Vert a\Vert _{S^{\bot }}\end{eqnarray}$$

for $a,b\in \mathbb{Z}/p\mathbb{Z}$ and $k\in \mathbb{Z}$ ; also we trivially have

$$\begin{eqnarray}\Vert a\Vert _{S^{\bot }}\leqslant \Vert a\Vert _{(S^{\prime })^{\bot }}\end{eqnarray}$$

if $S\subset S^{\prime }$ and $a\in \mathbb{Z}/p\mathbb{Z}$ , or equivalently that $B(S^{\prime },\unicode[STIX]{x1D70C})\subset B(S,\unicode[STIX]{x1D70C})$ for $\unicode[STIX]{x1D70C}>0$ . We will frequently use these inequalities in the sequel, usually without further comment. In Lemma 4.6 below, we will show that $\Vert \Vert _{S^{\bot }}$ is “dual” to a certain word norm $\Vert \Vert _{S}$ on $\mathbb{Z}/p\mathbb{Z}$ . One could also define Bohr sets in the case when $S$ is degenerate, but this creates some minor complications in our arguments, so we remove this case from our definition of a Bohr set.

We have the following standard size bounds for Bohr sets, whose proof may be found in [Reference Tao and Vu33, Lemma 4.20].

In previous work on Roth-type theorems, one sometimes restricts attention to regular Bohr sets, as first introduced in [Reference Bourgain6]; see [Reference Tao and Vu33, §4.4] for some discussion of this concept. Due to our use of the probabilistic method, we will be able to work with a technically simpler and “smoothed out” version of a regular Bohr set, which we call the regular probability distribution on a Bohr set.

Definition 4.3. Let $B(S,\unicode[STIX]{x1D70C})$ be a Bohr set. The regular probability distribution $\mathfrak{p}_{B(S,\unicode[STIX]{x1D70C})}:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{R}$ associated to $B(S,\unicode[STIX]{x1D70C})$ is the function defined by the formula

(4.2) $$\begin{eqnarray}\mathfrak{p}_{B(S,\unicode[STIX]{x1D70C})}(a):=2\int _{1/2}^{1}\frac{1_{B(S,t\unicode[STIX]{x1D70C})}(a)}{|B(S,t\unicode[STIX]{x1D70C})|}\,dt;\end{eqnarray}$$

it is easy to see (from Fubini’s theorem) that this is indeed a probability distribution on $\mathbb{Z}/p\mathbb{Z}$ . A random variable $\mathbf{a}\in \mathbb{Z}/p\mathbb{Z}$ is said to be drawn regularly from $B(S,\unicode[STIX]{x1D70C})$ if it has probability density function $\mathfrak{p}_{B(S,\unicode[STIX]{x1D70C})}$ , thus $\mathbb{P}(\mathbf{a}=a)=\mathfrak{p}_{B(S,\unicode[STIX]{x1D70C})}(a)$ for all $a\in \mathbb{Z}/p\mathbb{Z}$ .

More generally, for any shifted Bohr set $n+B(S,\unicode[STIX]{x1D70C})$ , we define the regular probability distribution $\mathfrak{p}_{n+B(S,\unicode[STIX]{x1D70C})}:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{R}$ by the formula

$$\begin{eqnarray}\mathfrak{p}_{n+B(S,\unicode[STIX]{x1D70C})}(a):=\mathfrak{p}_{B(S,\unicode[STIX]{x1D70C})}(a-n),\end{eqnarray}$$

and say that $\mathbf{a}$ is drawn regularly from $n+B(S,\unicode[STIX]{x1D70C})$ if it has probability distribution $\mathfrak{p}_{n+B(S,\unicode[STIX]{x1D70C})}$ .

Informally, to draw a random variable $\mathbf{a}$ regularly from $n+B(S,\unicode[STIX]{x1D70C})$ , one should draw it uniformly from $n\,+\,B(S,\mathbf{t}\unicode[STIX]{x1D70C})$ , where $\mathbf{t}$ is itself selected uniformly at random from the interval $[1/2,1]$ . Note that if $\mathbf{a}$ is drawn regularly from $n+B(S,\unicode[STIX]{x1D70C})$ , then $m+\mathbf{a}$ will be drawn regularly from $m+n+B(S,\unicode[STIX]{x1D70C})$ for any $m\in \mathbb{Z}/p\mathbb{Z}$ , and similarly $k\mathbf{a}$ will be drawn from $kn+B(k^{-1}\cdot S,\unicode[STIX]{x1D70C})$ for any non-zero $k\in \mathbb{Z}/p\mathbb{Z}$ , where $k^{-1}\cdot S:=\{k^{-1}\unicode[STIX]{x1D709}:\unicode[STIX]{x1D709}\in S\}$ is the dilate of the frequency set $S$ by $k^{-1}$ .

From Lemma 4.2 we see that if $\mathbf{a}$ is drawn regularly from a shifted Bohr set $n+B(S,\unicode[STIX]{x1D70C})$ , then

(4.3) $$\begin{eqnarray}\mathbb{P}(\mathbf{a}=a)\leqslant \frac{1}{(\unicode[STIX]{x1D70C}/2)^{|S|}p}\end{eqnarray}$$

for all $a\in \mathbb{Z}/p\mathbb{Z}$ . In practice, this will mean that the influence of any given value of $\mathbf{a}$ will be negligible.

The presence of the averaging parameter $t$ in (4.2) allows for the following very convenient approximate translation-invariance property. Given two random variables $\mathbf{a},\mathbf{a}^{\prime }$ taking values in a finite set $A$ , we define the total variation distance between the two to be the quantity

$$\begin{eqnarray}d_{\operatorname{TV}}(\mathbf{a},\mathbf{a}^{\prime }):=\mathop{\sum }_{a\in A}|\mathbb{P}(\mathbf{a}=a)-\mathbb{P}(\mathbf{a}^{\prime }=a)|,\end{eqnarray}$$

or equivalently

$$\begin{eqnarray}d_{\operatorname{TV}}(\mathbf{a},\mathbf{a}^{\prime })=\sup _{f}|\mathbb{E}f(\mathbf{a})-\mathbb{E}f(\mathbf{a}^{\prime })|\end{eqnarray}$$

where $f:A\rightarrow \mathbb{C}$ ranges over $1$ -bounded functions.

The next lemma gives some approximate translation-invariance properties of Bohr sets. Its proof is a thinly disguised version of the arguments of Bourgain [Reference Bourgain6].

Proof. To prove (i), it suffices to show that

$$\begin{eqnarray}\mathbb{E}f(\mathbf{a}+h)=\mathbb{E}f(\mathbf{a})+O\biggl(|S|\frac{\unicode[STIX]{x1D70C}^{\prime }}{\unicode[STIX]{x1D70C}}\biggr)\end{eqnarray}$$

for any $1$ -bounded function $f:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{C}$ ; the claim (ii) then also follows by conditioning $\mathbf{h}$ to a fixed value $h\in B(S^{\prime },\unicode[STIX]{x1D70C}^{\prime })$ , then multiplying by $\mathbb{P}(\mathbf{h}=h)$ and summing over $h$ .

By translating $f$ by $n$ , we may assume that $n=0$ . We may assume that $\unicode[STIX]{x1D70C}^{\prime }\leqslant \unicode[STIX]{x1D70C}/10|S|$ , as the claim is trivial otherwise.

From (4.2) we have

$$\begin{eqnarray}\mathbb{E}f(\mathbf{a})=2\int _{1/2}^{1}\mathop{\sum }_{a\in \mathbb{Z}/p\mathbb{Z}}f(a)\frac{1_{B(S,t\unicode[STIX]{x1D70C})}(a)}{|B(S,t\unicode[STIX]{x1D70C})|}\,dt\end{eqnarray}$$

and

$$\begin{eqnarray}\mathbb{E}f(\mathbf{a}+h)=2\int _{1/2}^{1}\mathop{\sum }_{a\in \mathbb{Z}/p\mathbb{Z}}f(a)\frac{1_{B(S,t\unicode[STIX]{x1D70C})-h}(a)}{|B(S,t\unicode[STIX]{x1D70C})|}\,dt\end{eqnarray}$$

so by the triangle inequality it suffices to show that

(4.4) $$\begin{eqnarray}\int _{1/2}^{1}\frac{\mathop{\sum }_{a\in \mathbb{Z}/p\mathbb{Z}}|1_{B(S,t\unicode[STIX]{x1D70C})}(a)-1_{B(S,t\unicode[STIX]{x1D70C})-h}(a)|}{|B(S,t\unicode[STIX]{x1D70C})|}\,dt\ll |S|\frac{\unicode[STIX]{x1D70C}^{\prime }}{\unicode[STIX]{x1D70C}}.\end{eqnarray}$$

By the triangle inequality, the integrand here is bounded above by $2$ . Also, from (4.1), we see that any $a$ for which $1_{B(S,t\unicode[STIX]{x1D70C})-h}(a)\neq 1_{B(S,t\unicode[STIX]{x1D70C})}(a)$ lies in the “annulus” $B(S,t\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D70C}^{\prime })\backslash B(S,t\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}^{\prime })$ . We conclude that the left-hand side of (4.4) is bounded by

$$\begin{eqnarray}\int _{1/2}^{1}O\biggl(\min \biggl(\frac{|B(S,t\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D70C}^{\prime })|-|B(S,t\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}^{\prime })|}{|B(S,t\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}^{\prime })|},1\biggr)\biggr)\,dt\end{eqnarray}$$

which, using the elementary bound $\min (x-1,1)\ll \log x$ for $x\geqslant 1$ , can be bounded in turn by

$$\begin{eqnarray}O\biggl(\int _{1/2}^{1}\log \frac{|B(S,t\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D70C}^{\prime })|}{|B(S,t\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}^{\prime })|}\,dt\biggr).\end{eqnarray}$$

The integral telescopes to

$$\begin{eqnarray}O\biggl(\int _{1}^{1+\unicode[STIX]{x1D70C}^{\prime }/\unicode[STIX]{x1D70C}}\log |B(S,t\unicode[STIX]{x1D70C})|\,dt-\int _{1/2-\unicode[STIX]{x1D70C}^{\prime }/\unicode[STIX]{x1D70C}}^{1/2}\log |B(S,t\unicode[STIX]{x1D70C})|\,dt\biggr)\end{eqnarray}$$

which can be bounded in turn by

$$\begin{eqnarray}O\biggl(\frac{\unicode[STIX]{x1D70C}^{\prime }}{\unicode[STIX]{x1D70C}}\log \frac{|B(S,2\unicode[STIX]{x1D70C})|}{|B(S,\unicode[STIX]{x1D70C}/4)|}\biggr).\end{eqnarray}$$

The claim now follows from Lemma 4.2. ◻

We will be interested in the Fourier coefficients $\mathbb{E}e_{p}(\unicode[STIX]{x1D706}\mathbf{n})=\mathbb{E}e(\unicode[STIX]{x1D706}\mathbf{n}/p)$ of random variables $\mathbf{n}$ drawn regularly from Bohr sets $B(S,\unicode[STIX]{x1D70C})$ . As was noted by Bourgain [Reference Bourgain6], these coefficients are controlled by a “word norm” $\Vert \Vert _{S}$ , defined as follows.

Similarly to (4.1), we observe the triangle inequalities

(4.5) $$\begin{eqnarray}\Vert a+b\Vert _{S}\leqslant \Vert a\Vert _{S}+\Vert b\Vert _{S};\qquad \Vert ka\Vert _{S}\leqslant |k|\Vert a\Vert _{S}\end{eqnarray}$$

for $a,b\in \mathbb{Z}/p\mathbb{Z}$ and $k\in \mathbb{Z}$ , which we will use frequently in the sequel, often without further comment.

We now give a duality relationship between the word norm $\Vert \Vert _{S}$ and the dual $S$ -norm $\Vert \Vert _{S^{\bot }}$ .

Proof. To prove (i), we simply observe (using (2.4)) that for any $n\in \mathbb{Z}/p\mathbb{Z}$ , one has

$$\begin{eqnarray}\displaystyle & & \displaystyle \Vert n\unicode[STIX]{x1D706}/p\Vert _{\mathbb{R}/\mathbb{Z}}\nonumber\\ \displaystyle & & \displaystyle \quad =\biggl\|\mathop{\sum }_{\unicode[STIX]{x1D709}\in S}a_{\unicode[STIX]{x1D709}}\frac{n\unicode[STIX]{x1D709}}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}\leqslant \mathop{\sum }_{\unicode[STIX]{x1D709}\in S}|a_{\unicode[STIX]{x1D709}}|\biggl\|\frac{n\unicode[STIX]{x1D709}}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}\leqslant \mathop{\sum }_{\unicode[STIX]{x1D709}\in S}|a_{\unicode[STIX]{x1D709}}|\Vert n\Vert _{S^{\bot }}\leqslant \Vert \unicode[STIX]{x1D706}\Vert _{S}\Vert n\Vert _{S^{\bot }}\nonumber\end{eqnarray}$$

as desired, where $\unicode[STIX]{x1D706}=\sum _{\unicode[STIX]{x1D709}\in S}a_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D709}$ is a representation of $\unicode[STIX]{x1D706}$ that minimizes $\sum _{\unicode[STIX]{x1D709}\in S}|\unicode[STIX]{x1D709}|$ .

Estimates such as (ii) go back to the work of Bourgain [Reference Bourgain6]. We will prove this claim by a Fourier-analytic argument. We may assume that $\Vert \unicode[STIX]{x1D706}\Vert _{S}\geqslant |S|^{3/2}$ , as the claim is trivial otherwise. Let $\unicode[STIX]{x1D713}:\mathbb{R}\rightarrow \mathbb{R}$ be a non-negative smooth even function (not depending on $p$ or $\unicode[STIX]{x1D706}$ ) supported on $[-1,1]$ and non-zero on $[-1/2,1/2]$ , whose Fourier transform $\hat{\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D709}):=\int _{\mathbb{R}}\unicode[STIX]{x1D713}(x)e(-\unicode[STIX]{x1D709}x)\,dx$ is also non-negative. Set $N:=|S|^{-1}\Vert \unicode[STIX]{x1D706}\Vert _{S}$ , so in particular $N\geqslant 1$ . We consider the kernel $K_{N}:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{C}$ defined by

$$\begin{eqnarray}K_{N}(n):=\mathop{\sum }_{k\in \mathbb{Z}}e_{p}(kn)\unicode[STIX]{x1D713}\biggl(\frac{k}{N}\biggr);\end{eqnarray}$$

by the Poisson summation formula we have

$$\begin{eqnarray}K_{N}(n~(\operatorname{mod}~p))=N\mathop{\sum }_{m\in \mathbb{Z}}\hat{\unicode[STIX]{x1D713}}\biggl(\frac{Nn}{p}-Nm\biggr)\end{eqnarray}$$

for any integer $n$ , so in particular $K_{N}$ is non-negative.

By definition of $N$ , the frequency $\unicode[STIX]{x1D706}$ has no representations of the form $\unicode[STIX]{x1D706}=\sum _{\unicode[STIX]{x1D709}\in S}a_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D709}$ with $\sup _{\unicode[STIX]{x1D709}\in S}|a_{\unicode[STIX]{x1D709}}|<N$ . Hence the Riesz-type product $\prod _{\unicode[STIX]{x1D709}\in S}K_{N}(\unicode[STIX]{x1D709}n)$ , when expanded, contains no terms of the form $e_{p}(\unicode[STIX]{x1D706}n)$ or $e_{p}(-\unicode[STIX]{x1D706}n)$ , and is therefore orthogonal to $\cos (2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D706}n/p)$ . In particular we have the identity

$$\begin{eqnarray}\mathbb{E}_{n\in \mathbb{Z}/p\mathbb{Z}}\mathop{\prod }_{\unicode[STIX]{x1D709}\in S}K_{N}(\unicode[STIX]{x1D709}n)=\mathbb{E}_{n\in \mathbb{Z}/p\mathbb{Z}}\biggl(1-\cos \biggl(\frac{2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D706}n}{p}\biggr)\biggr)\mathop{\prod }_{\unicode[STIX]{x1D709}\in S}K_{N}(\unicode[STIX]{x1D709}n).\end{eqnarray}$$

On the other hand, from two applications of (2.3) we have

$$\begin{eqnarray}\displaystyle 1-\cos \biggl(\frac{2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D706}n}{p}\biggr) & \ll & \displaystyle \biggl\|\frac{\unicode[STIX]{x1D706}n}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}^{2}\leqslant A^{2}\Vert n\Vert _{S^{\bot }}^{2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle A^{2}\mathop{\sum }_{\unicode[STIX]{x1D709}_{0}\in S}\biggl\|\frac{\unicode[STIX]{x1D709}_{0}n}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}^{2}\leqslant A^{2}\mathop{\sum }_{\unicode[STIX]{x1D709}_{0}\in S}\biggl(1-\cos \biggl(\frac{2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D709}_{0}n}{p}\biggr)\biggr).\nonumber\end{eqnarray}$$

As $K_{N}$ is non-negative, we conclude that

(4.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{E}_{n\in \mathbb{Z}/p\mathbb{Z}}\mathop{\prod }_{\unicode[STIX]{x1D709}\in S}K_{N}(\unicode[STIX]{x1D709}n)\nonumber\\ \displaystyle & & \displaystyle \quad \ll A^{2}\mathop{\sum }_{\unicode[STIX]{x1D709}_{0}\in S}\mathbb{E}_{n\in \mathbb{Z}/p\mathbb{Z}}\biggl(\biggl(\mathop{\prod }_{\unicode[STIX]{x1D709}\in S\backslash \unicode[STIX]{x1D709}_{0}}\!K_{\!N}(\unicode[STIX]{x1D709}n)\biggr)K_{\!N}(\unicode[STIX]{x1D709}_{0}n)\biggl(1-\cos \biggl(\frac{2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D709}_{0}n}{p}\biggr)\biggr)\biggr).\end{eqnarray}$$

We can expand $K_{N}(\unicode[STIX]{x1D709}_{0}n)(1-\cos (2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D709}_{0}n/p))$ as a Fourier series

$$\begin{eqnarray}\mathop{\sum }_{k\in \mathbb{Z}}e_{p}(kn)\biggl(\unicode[STIX]{x1D713}\biggl(\frac{k}{N}\biggr)-\frac{\unicode[STIX]{x1D713}((k-1)/N)+\unicode[STIX]{x1D713}((k+1)/N)}{2}\biggr).\end{eqnarray}$$

The expression inside parentheses is only non-vanishing for $|k|\leqslant N+1$ , and has magnitude $O(1/N^{2})$ . As $\unicode[STIX]{x1D713}$ is non-negative everywhere and non-zero on $[-1/2,1/2]$ , we thus have a pointwise estimate of the form

$$\begin{eqnarray}\unicode[STIX]{x1D713}\biggl(\frac{k}{N}\biggr)-\frac{\unicode[STIX]{x1D713}((k-1)/N)+\unicode[STIX]{x1D713}((k+1)/N)}{2}\ll \frac{1}{N^{2}}\mathop{\sum }_{j=-8}^{8}\unicode[STIX]{x1D713}\biggl(\frac{k}{N}-\frac{j}{4}\biggr)\end{eqnarray}$$

(for example). By using the non-negativity of the Fourier coefficients of $K_{N}$ , this gives the estimate

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{E}_{n\in \mathbb{Z}/p\mathbb{Z}}\biggl(\mathop{\prod }_{\unicode[STIX]{x1D709}\in S\backslash \unicode[STIX]{x1D709}_{0}}K_{N}(\unicode[STIX]{x1D709}n)\biggr)K_{N}(\unicode[STIX]{x1D709}_{0}n)\biggl(1-\cos \biggl(\frac{2\unicode[STIX]{x1D70B}\unicode[STIX]{x1D709}_{0}n}{p}\biggr)\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad \ll \frac{1}{N^{2}}\mathbb{E}_{n\in \mathbb{Z}/p\mathbb{Z}}\mathop{\prod }_{\unicode[STIX]{x1D709}\in S}K_{N}(\unicode[STIX]{x1D709}n).\nonumber\end{eqnarray}$$

Comparing this with (4.6), we conclude that $1\ll A^{2}|S|/N^{2}$ , and the claim follows from the definition of $N$ .◻

Next, we estimate the Fourier coefficients of a regular distribution on a Bohr set in terms of the word norm.

Lemma 4.7. Let $S$ be a non-degenerate subset of $\mathbb{Z}/p\mathbb{Z}$ . Suppose that $\mathbf{n}$ is drawn regularly from $B(S,\unicode[STIX]{x1D70C})$ . Then we have

$$\begin{eqnarray}\mathbb{E}e_{p}(\unicode[STIX]{x1D706}\mathbf{n})\ll \frac{|S|^{5/2}}{\unicode[STIX]{x1D70C}\Vert \unicode[STIX]{x1D706}\Vert _{S}}\end{eqnarray}$$

for all $\unicode[STIX]{x1D706}\in \mathbb{Z}/p\mathbb{Z}$ , where we adopt the convention that the above estimate is vacuously true if $\Vert \unicode[STIX]{x1D706}\Vert _{S}=0$ .

Proof. For any $h\in \mathbb{Z}/p\mathbb{Z}$ , one has from Lemma 4.4 that

$$\begin{eqnarray}\mathbb{E}e_{p}(\unicode[STIX]{x1D706}\mathbf{n})=\mathbb{E}e_{p}(\unicode[STIX]{x1D706}(\mathbf{n}+h))+O\biggl(\frac{|S|\Vert h\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}}\biggr)\end{eqnarray}$$

which we may rearrange as

$$\begin{eqnarray}(1-e_{p}(\unicode[STIX]{x1D706}h))\mathbb{E}e_{p}(\unicode[STIX]{x1D706}\mathbf{n})\ll \frac{|S|\Vert h\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}}.\end{eqnarray}$$

Since $|1-e_{p}(\unicode[STIX]{x1D706}h)|\gg \Vert \unicode[STIX]{x1D706}h/p\Vert _{\mathbb{R}/\mathbb{Z}}$ , we conclude that

$$\begin{eqnarray}\biggl\|\frac{\unicode[STIX]{x1D706}h}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}\mathbb{E}e_{p}(\unicode[STIX]{x1D706}\mathbf{n})\ll \frac{|S|\Vert h\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}}.\end{eqnarray}$$

Taking $h$ so as to minimize the ratio $\Vert h\Vert _{S^{\bot }}/\Vert \unicode[STIX]{x1D706}h/p\Vert _{\mathbb{R}/\mathbb{Z}}$ , the claim follows from Lemma 4.6.◻

We will take advantage of the fact that Bohr sets can be approximately described as generalized arithmetic progressions. A key lemma in this regard is the following.

Lemma 4.8. Let $\unicode[STIX]{x1D6E4}$ be a lattice in $\mathbb{R}^{d}$ . Then there exist linearly independent generators $v_{1},\ldots ,v_{d}$ of $\unicode[STIX]{x1D6E4}$ and real numbers $N_{1},\ldots ,N_{d}>0$ such that

(4.7) $$\begin{eqnarray}B_{\mathbb{R}^{d}}(0,O(d)^{-3d/2}t)\cap \unicode[STIX]{x1D6E4}\subset \biggl\{\mathop{\sum }_{i=1}^{d}n_{i}v_{i}:|n_{i}|<tN_{i}\biggr\}\subset B_{\mathbb{R}^{d}}(0,t)\cap \unicode[STIX]{x1D6E4}\end{eqnarray}$$

for all $t>0$ , where $B_{\mathbb{R}^{d}}(0,r)$ is the open Euclidean ball of radius $r$ in $\mathbb{R}^{d}$ , and the $n_{i}$ are understood to be integers. Furthermore, the determinant/covolume $\det (\unicode[STIX]{x1D6E4})$ obeys the bounds

(4.8) $$\begin{eqnarray}\det (\unicode[STIX]{x1D6E4})=(2d)^{O(d)}\mathop{\prod }_{i=1}^{d}N_{i}^{-1}.\end{eqnarray}$$

Proof. Applying [Reference Tao and Vu34, Theorem 1.6], we can find elements $v_{1},\ldots ,v_{r}$ of $\unicode[STIX]{x1D6E4}$ for some $r\leqslant d$ , linearly independent over the rationals, and real numbers $N_{1},\ldots ,N_{d}>0$ such that

(4.9) $$\begin{eqnarray}B_{\mathbb{R}^{d}}(0,O(d)^{-3d/2}t)\cap \unicode[STIX]{x1D6E4}\subset \biggl\{\mathop{\sum }_{i=1}^{r}n_{i}v_{i}:|n_{i}|<tN_{i}\biggr\}\subset B_{\mathbb{R}^{d}}(0,t)\cap \unicode[STIX]{x1D6E4}\end{eqnarray}$$

for all $t>0$ , and such that

$$\begin{eqnarray}O(d)^{-7d/2}|B_{\mathbb{ R}^{d}}(0,t)\cap \unicode[STIX]{x1D6E4}|\leqslant \biggl|\biggl\{\mathop{\sum }_{i=1}^{r}n_{i}v_{i}:|n_{i}|<tN_{i}\biggr\}\biggr|\leqslant |B_{\mathbb{R}^{d}}(0,t)\cap \unicode[STIX]{x1D6E4}|.\end{eqnarray}$$

(Strictly speaking, the statement of [Reference Tao and Vu34, Theorem 1.6] only claims the latter bound for $t=1$ , but the same argument gives the bound for all $t>0$ .) Sending $t$ to infinity, we conclude that the $v_{1},\ldots ,v_{r}$ generate $\unicode[STIX]{x1D6E4}$ ; since, by virtue of being a lattice, $\unicode[STIX]{x1D6E4}$ is cocompact, this forces $d=r$ . Also, volume packing arguments show that as $t\rightarrow \infty$ , the cardinality $|B_{\mathbb{R}^{d}}(0,t)\,\cap \,\unicode[STIX]{x1D6E4}|$ is asymptotic to the measure of $B_{\mathbb{R}^{d}}(0,t)$ divided by $\det (\unicode[STIX]{x1D6E4})$ , while the cardinality of $|\{n_{1}v_{1}+\cdots +n_{d}v_{d}:|n_{i}|\leqslant tN_{i}\}|$ is asymptotic to $\prod _{i=1}^{d}(2tN_{i})$ . We conclude (4.8) as desired.◻

The following corollary describes how we may pick a “basis” for a Bohr set.

Corollary 4.9. Let $S$ be a non-degenerate subset of $\mathbb{Z}/p\mathbb{Z}$ , and set $d:=|S|$ . Then there exist elements $a_{1},\ldots ,a_{d}$ of $\mathbb{Z}/p\mathbb{Z}$ and real numbers $N_{1},\ldots ,N_{d}>0$ such that

(4.10) $$\begin{eqnarray}\mathop{\prod }_{i=1}^{d}N_{i}^{-1}=(2d)^{O(d)}p\end{eqnarray}$$

and

(4.11) $$\begin{eqnarray}\Vert a_{i}\Vert _{S^{\bot }}\leqslant N_{i}^{-1}\end{eqnarray}$$

for all $i=1,\ldots ,d$ . Furthermore, for any $a\in \mathbb{Z}/p\mathbb{Z}$ , there exists a representation

(4.12) $$\begin{eqnarray}a=n_{1}a_{1}+\cdots +n_{d}a_{d}\end{eqnarray}$$

with $n_{1},\ldots ,n_{d}$ integers of size

(4.13) $$\begin{eqnarray}n_{i}=(2d)^{O(d)}N_{i}\Vert a\Vert _{S^{\bot }}\end{eqnarray}$$

for $i=1,\ldots ,d$ . Finally, if one imposes the additional condition $|n_{i}|<N_{i}/2$ for all $i=1,\ldots ,d$ , then there is at most one such representation of this form (4.12) for a given $a$ .

Proof. For each $s\in S$ , the fraction $s/p$ can be viewed as an element of $\mathbb{R}/\mathbb{Z}$ of order at most $p$ ; as $S$ is non-degenerate, we see that the tuple $(s/p)_{s\in S}$ is an element of the torus $(\mathbb{R}/\mathbb{Z})^{S}$ of order $p$ . Let $\unicode[STIX]{x1D6E4}$ be the preimage in $\mathbb{R}^{S}$ of the group generated by this element, thus $\unicode[STIX]{x1D6E4}$ is a lattice of $\mathbb{R}^{S}$ that contains $\mathbb{Z}^{S}$ as a sublattice of index $p$ ; in particular, $\unicode[STIX]{x1D6E4}$ has determinant $p$ . Applying Lemma 4.8, one can find generators $v_{1},\ldots ,v_{d}$ of $\unicode[STIX]{x1D6E4}$ and real numbers $N_{1},\ldots ,N_{d}$ obeying (4.10) such that

(4.14) $$\begin{eqnarray}B_{\mathbb{R}^{S}}(0,O(d)^{-3d/2}t)\cap \unicode[STIX]{x1D6E4}\subset \biggl\{\mathop{\sum }_{i=1}^{d}n_{i}v_{i}:|n_{i}|<tN_{i}\biggr\}\subset B_{\mathbb{R}^{S}}(0,t)\cap \unicode[STIX]{x1D6E4}\end{eqnarray}$$

for all $t>0$ .

By construction of $\unicode[STIX]{x1D6E4}$ , we can find elements $a_{1},\ldots ,a_{d}$ of $\mathbb{Z}/p\mathbb{Z}$ such that

(4.15) $$\begin{eqnarray}v_{i}=\biggl(\frac{a_{i}s}{p}\biggr)_{s\in S}\quad (\operatorname{mod}~\mathbb{Z}^{S})\end{eqnarray}$$

for $i=1,\ldots ,d$ . Applying (4.14) with $t$ slightly larger than $N_{i}^{-1}$ for some $i=1,\ldots ,d$ , we see that $v_{i}\in B_{\mathbb{R}^{d}}(N_{i}^{-1})$ , and hence by (4.15) we have (4.11).

Finally, if $a\in \mathbb{Z}/p\mathbb{Z}$ , then by definition of $\unicode[STIX]{x1D6E4}$ we can find an element $x$ of $\unicode[STIX]{x1D6E4}$ in the preimage of $(as/p)_{s\in S}$ such that each component of $x$ has magnitude less than $\Vert a\Vert _{S^{\bot }}$ ; in particular, $x\in B_{\mathbb{R}^{S}}(0,\sqrt{d}\Vert a\Vert _{S^{\bot }})$ . Applying (4.14), we conclude that $x=\sum _{i=1}^{d}n_{i}v_{i}$ for some integers $n_{1},\ldots ,n_{d}$ obeying (4.13), giving the desired representation (4.12).

Finally, we show uniqueness. If there were two representations of the form (4.12) with $|n_{i}|<N_{i}/2$ for all $i=1,\ldots ,d$ , then there exists a tuple $(n_{1}^{\prime },\ldots ,n_{d}^{\prime })\in \mathbb{Z}^{d}$ , not identically zero, with $|n_{i}^{\prime }|<N_{i}$ for all $i=1,\ldots ,d$ and $\sum _{i=1}^{d}n_{i}a_{i}=0$ , which implies that the vector $\sum _{i=1}^{d}n_{i}v_{i}$ lies in $\mathbb{Z}^{S}$ . As the $v_{1},\ldots ,v_{d}$ are linearly independent, this vector must have magnitude at least $1$ ; but this contradicts (4.7) (with $t=1$ ).◻

Linear and quadratic functions on Bohr sets

We will frequently need to deal with locally linear or quadratic functions on Bohr sets. We review the definitions of these now.

Definition 4.10. Let $B$ be a subset of $\mathbb{Z}/p\mathbb{Z}$ , and let $G=(G,+)$ be an abelian group. A function $\unicode[STIX]{x1D719}:B\rightarrow G$ is said to be locally linear on $B$ if one has

$$\begin{eqnarray}\unicode[STIX]{x1D719}(n+h_{1}+h_{2})-\unicode[STIX]{x1D719}(n+h_{1})-\unicode[STIX]{x1D719}(n+h_{2})+\unicode[STIX]{x1D719}(n)=0\end{eqnarray}$$

whenever $n,h_{1},h_{2}\in \mathbb{Z}/p\mathbb{Z}$ are such that $n,n+h_{1},n+h_{2},n+h_{1}+h_{2}\in B$ . Similarly, $\unicode[STIX]{x1D719}$ is said to be locally quadratic on $B$ if one has

(4.16) $$\begin{eqnarray}\mathop{\sum }_{(\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3})\in \{0,1\}^{3}}(-1)^{\unicode[STIX]{x1D714}_{1}+\unicode[STIX]{x1D714}_{2}+\unicode[STIX]{x1D714}_{3}}\unicode[STIX]{x1D719}(n+\unicode[STIX]{x1D714}_{1}h_{1}+\unicode[STIX]{x1D714}_{2}h_{2}+\unicode[STIX]{x1D714}_{3}h_{3})=0\end{eqnarray}$$

whenever $n,h_{1},h_{2},h_{3}\in \mathbb{Z}/p\mathbb{Z}$ are such that $n+\unicode[STIX]{x1D714}_{1}h_{1}+\unicode[STIX]{x1D714}_{2}h_{2}+\unicode[STIX]{x1D714}_{3}h_{3}\in B$ for all $(\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3})\in \{0,1\}^{3}$ .

A function $\unicode[STIX]{x1D713}:B\times B\rightarrow G$ is said to be locally bilinear on $B$ if one has

$$\begin{eqnarray}\unicode[STIX]{x1D713}(h_{1}+h_{1}^{\prime },h_{2})=\unicode[STIX]{x1D713}(h_{1},h_{2})+\unicode[STIX]{x1D713}(h_{1}^{\prime },h_{2})\end{eqnarray}$$

whenever $h_{1},h_{1}^{\prime },h_{2}\in B$ are such that $h_{1}+h_{1}^{\prime }\in B$ , and similarly one has

$$\begin{eqnarray}\unicode[STIX]{x1D713}(h_{1},h_{2}+h_{2}^{\prime })=\unicode[STIX]{x1D713}(h_{1},h_{2})+\unicode[STIX]{x1D713}(h_{1},h_{2}^{\prime })\end{eqnarray}$$

whenever $h_{1},h_{2},h_{2}^{\prime }\in B$ are such that $h_{2}+h_{2}^{\prime }\in B$ .

Specializing (4.16) to the case $h_{1}=h_{2}=h_{3}=h$ , we conclude that

(4.17) $$\begin{eqnarray}\unicode[STIX]{x1D719}(n)-3\unicode[STIX]{x1D719}(n+h)+3\unicode[STIX]{x1D719}(n+2h)-\unicode[STIX]{x1D719}(n+3h)=0\end{eqnarray}$$

whenever $\unicode[STIX]{x1D719}:B\rightarrow G$ is locally quadratic on $B$ and $n,n+h,n+2h,n+3h\in B$ .

It is well known (from the Weyl exponential sum estimates) that quadratic exponential sums such as $\mathbb{E}_{1\leqslant n\leqslant N}e(\unicode[STIX]{x1D6FC}n^{2}+\unicode[STIX]{x1D6FD}n)$ can only be large when the quadratic phase $\unicode[STIX]{x1D6FC}n^{2}$ is of “major arc” type in the sense that $k\unicode[STIX]{x1D6FC}n^{2}$ is close to constant on the range $\{1,\ldots ,N\}$ of the summation variable $n$ , for some bounded positive integer $k$ . The following proposition is an analogue of this phenomenon on Bohr sets.

Proposition 4.11 (Large local quadratic exponential sums).

Let $B(S,\unicode[STIX]{x1D70C})$ be a Bohr set, let $0<\unicode[STIX]{x1D6FF}\leqslant 1/2$ , let $\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}:B(S,10\unicode[STIX]{x1D70C})\rightarrow \mathbb{R}/\mathbb{Z}$ be locally linear maps, and let $\unicode[STIX]{x1D719}:B(S,10\unicode[STIX]{x1D70C})\times B(S,10\unicode[STIX]{x1D70C})\rightarrow \mathbb{R}/\mathbb{Z}$ be a locally bilinear phase such that

(4.18) $$\begin{eqnarray}|\mathbb{E}e(\unicode[STIX]{x1D719}(\mathbf{n},\mathbf{m})+\unicode[STIX]{x1D706}(\mathbf{n})+\unicode[STIX]{x1D707}(\mathbf{m}))|\geqslant \unicode[STIX]{x1D6FF}\end{eqnarray}$$

if $\mathbf{n},\mathbf{m}$ are drawn independently and regularly from $B(S,\unicode[STIX]{x1D70C})$ . Then there exists a natural number

$$\begin{eqnarray}1\leqslant k\leqslant \unicode[STIX]{x1D6FF}^{-O(C_{1}|S|^{2})}\end{eqnarray}$$

such that

(4.19) $$\begin{eqnarray}\Vert k\unicode[STIX]{x1D719}(n,m)\Vert _{\mathbb{R}/\mathbb{Z}}\ll \unicode[STIX]{x1D6FF}^{-O(C_{1}|S|^{2})}\frac{\Vert n\Vert _{S}\Vert m\Vert _{S}}{\unicode[STIX]{x1D70C}^{2}}\end{eqnarray}$$

whenever $n,m\in B(S,\unicode[STIX]{x1D6FF}^{C_{1}}\unicode[STIX]{x1D70C}/(C_{1}|S|)^{3|S|})$ .

Proof. Let $d:=|S|$ , thus $d\geqslant 1$ . By Corollary 4.9, we can find elements $a_{1},\ldots ,a_{d}$ of $\mathbb{Z}/p\mathbb{Z}$ and real numbers $N_{1},\ldots ,N_{d}$ obeying the conclusions of that corollary.

Suppose that $1\leqslant i,j\leqslant d$ are such that $N_{i},N_{j}\geqslant d/\unicode[STIX]{x1D6FF}^{C_{1}/2}\unicode[STIX]{x1D70C}$ (we allow $i$ and $j$ to be equal). Then by (4.11) we have

$$\begin{eqnarray}\Vert a_{i}\Vert _{S^{\bot }},\Vert a_{j}\Vert _{S^{\bot }}\leqslant d^{-1}\unicode[STIX]{x1D6FF}^{C_{1}/2}\unicode[STIX]{x1D70C}.\end{eqnarray}$$

We can control the coefficient $\unicode[STIX]{x1D719}(a_{i},a_{j})$ by the following argument. If we draw $\mathbf{b}_{i}$ and $\mathbf{b}_{j}$ uniformly from $\{b_{i}\in \mathbb{Z}:1\leqslant b_{i}\leqslant \unicode[STIX]{x1D6FF}^{C_{1}/4}N_{i}\unicode[STIX]{x1D70C}/d\}$ and $\{b_{j}\in \mathbb{Z}:1\leqslant b_{j}\leqslant \unicode[STIX]{x1D6FF}^{C_{1}/4}N_{j}\unicode[STIX]{x1D70C}/d\}$ respectively and independently of each other and of $\mathbf{n},\mathbf{m}$ , then from two applications of Lemma 4.4 (comparing $\mathbf{n}$ with $\mathbf{n}+\mathbf{b}_{i}a_{i}$ , and $\mathbf{m}$ with $\mathbf{m}+\mathbf{b}_{j}a_{j}$ ) we have

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{E}e(\unicode[STIX]{x1D719}(\mathbf{n}+\mathbf{b}_{i}a_{i},\mathbf{m}+\mathbf{b}_{j}a_{j})+\unicode[STIX]{x1D706}(\mathbf{n}+\mathbf{b}_{i}a_{i})+\unicode[STIX]{x1D707}(\mathbf{m}+\mathbf{b}_{j}a_{j}))\nonumber\\ \displaystyle & & \displaystyle \quad =\mathbb{E}e(\unicode[STIX]{x1D719}(\mathbf{n},\mathbf{m})+\unicode[STIX]{x1D706}(\mathbf{n})+\unicode[STIX]{x1D707}(\mathbf{m}))+O(\unicode[STIX]{x1D6FF}^{C_{1}/4})\nonumber\end{eqnarray}$$

and hence from (4.18) (assuming $C_{1}$ large enough) we have

$$\begin{eqnarray}|\mathbb{E}e(\unicode[STIX]{x1D719}(\mathbf{n}+\mathbf{b}_{i}a_{i},\mathbf{m}+\mathbf{b}_{j}a_{j})+\unicode[STIX]{x1D706}(\mathbf{n}+\mathbf{b}_{i}a_{i})+\unicode[STIX]{x1D707}(\mathbf{m}+\mathbf{b}_{j}a_{j}))|\gg \unicode[STIX]{x1D6FF}.\end{eqnarray}$$

By the pigeonhole principle, we can therefore find $n,m\in B(S,\unicode[STIX]{x1D70C})$ such that

$$\begin{eqnarray}|\mathbb{E}e(\unicode[STIX]{x1D719}(n+\mathbf{b}_{i}a_{i},m+\mathbf{b}_{j}a_{j})+\unicode[STIX]{x1D706}(n+\mathbf{b}_{i}a_{i})+\unicode[STIX]{x1D707}(m+\mathbf{b}_{j}a_{j}))|\gg \unicode[STIX]{x1D6FF}.\end{eqnarray}$$

Using the local bilinearity of $\unicode[STIX]{x1D719}$ , the left-hand side may be written as

$$\begin{eqnarray}|\mathbb{E}e(\mathbf{b}_{i}\mathbf{b}_{j}\unicode[STIX]{x1D719}(a_{i},a_{j})+\unicode[STIX]{x1D6FC}\mathbf{b}_{i}+\unicode[STIX]{x1D6FD}\mathbf{b}_{j}+\unicode[STIX]{x1D6FE})|\end{eqnarray}$$

for some $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE}\in \mathbb{R}/\mathbb{Z}$ depending on $i,j,n,m$ whose exact values are not of importance to us. Evaluating the expectations and using the triangle inequality, we conclude that

$$\begin{eqnarray}\mathbb{E}_{1\leqslant b_{i}\leqslant \unicode[STIX]{x1D6FF}^{C_{1}/4}N_{i}\unicode[STIX]{x1D70C}/d}|\mathbb{E}_{1\leqslant b_{j}\leqslant \unicode[STIX]{x1D6FF}^{C_{1}/4}N_{j}\unicode[STIX]{x1D70C}/d}e(b_{j}(b_{i}\unicode[STIX]{x1D719}(a_{i},a_{j})+\unicode[STIX]{x1D6FD}))|\gg \unicode[STIX]{x1D6FF}\end{eqnarray}$$

and hence (by Lemma 2.2)

$$\begin{eqnarray}|\mathbb{E}_{1\leqslant b_{j}\leqslant \unicode[STIX]{x1D6FF}^{C_{1}/4}N_{j}\unicode[STIX]{x1D70C}/d}e(b_{j}(b_{i}\unicode[STIX]{x1D719}(a_{i},a_{j})+\unicode[STIX]{x1D6FD}))|\gg \unicode[STIX]{x1D6FF}\end{eqnarray}$$

for $\gg \unicode[STIX]{x1D6FF}^{C_{1}/4+1}N_{i}\unicode[STIX]{x1D70C}/d$ values of $b_{i}$ in the range $1\leqslant b_{i}\leqslant \unicode[STIX]{x1D6FF}^{C_{1}/4}N_{i}\unicode[STIX]{x1D70C}/d$ . This average is a geometric series that can be explicitly computed, leading to the bound

$$\begin{eqnarray}\Vert b_{i}\unicode[STIX]{x1D719}(a_{i},a_{j})+\unicode[STIX]{x1D6FD}\Vert _{\mathbb{R}/\mathbb{Z}}\ll \frac{d}{\unicode[STIX]{x1D6FF}^{C_{1}/4+1}N_{j}\unicode[STIX]{x1D70C}}\end{eqnarray}$$

for $\gg \unicode[STIX]{x1D6FF}^{C_{1}/4+1}N_{i}\unicode[STIX]{x1D70C}/d$ values of $b_{i}$ in the range $1\leqslant b_{i}\leqslant \unicode[STIX]{x1D6FF}^{C_{1}/4}N_{i}\unicode[STIX]{x1D70C}/d$ . Applying [Reference Green and Tao17, Lemma A.4] (which is really an observation of Vinogradov, used often in the theory of Weyl sums), we conclude that

$$\begin{eqnarray}\Vert k_{i,j}\unicode[STIX]{x1D719}(a_{i},a_{j})\Vert _{\mathbb{R}/\mathbb{Z}}\ll \frac{d^{2}}{\unicode[STIX]{x1D6FF}^{O(C_{1})}N_{i}N_{j}\unicode[STIX]{x1D70C}^{2}}\end{eqnarray}$$

for some natural number $k_{i,j}$ with $1\leqslant k_{i,j}\ll \unicode[STIX]{x1D6FF}^{-O(C_{1})}$ . If we then “clear denominators” by defining

$$\begin{eqnarray}k:=\mathop{\prod }_{1\leqslant i,j\leqslant d:N_{i},N_{j}\geqslant d/\unicode[STIX]{x1D6FF}^{C_{1}/2}\unicode[STIX]{x1D70C}}k_{i,j},\end{eqnarray}$$

then $1\leqslant k\ll \unicode[STIX]{x1D6FF}^{-O(C_{1}d^{2})}$ and

(4.20) $$\begin{eqnarray}\Vert k\unicode[STIX]{x1D719}(a_{i},a_{j})\Vert _{\mathbb{R}/\mathbb{Z}}\ll \frac{1}{\unicode[STIX]{x1D6FF}^{O(C_{1}d^{2})}N_{i}N_{j}\unicode[STIX]{x1D70C}^{2}}\end{eqnarray}$$

for all $1\leqslant i,j\leqslant d$ with $N_{i},N_{j}\geqslant d/\unicode[STIX]{x1D6FF}^{C_{1}/2}\unicode[STIX]{x1D70C}$ .

For any $n\in \mathbb{Z}/p\mathbb{Z}$ , we see from Corollary 4.9 that we can find integers $n_{1},\ldots ,n_{d}$ with

$$\begin{eqnarray}n_{i}\ll (2d)^{O(d)}N_{i}\Vert n\Vert _{S^{\bot }}\end{eqnarray}$$

such that

$$\begin{eqnarray}n=n_{1}a_{1}+\cdots +n_{d}a_{d}.\end{eqnarray}$$

In particular, if $n\in B(S,\unicode[STIX]{x1D6FF}^{C_{1}}\unicode[STIX]{x1D70C}/(C_{1}d)^{3d})$ , then $n_{i}$ is only non-zero when $N_{i}\geqslant d/\unicode[STIX]{x1D6FF}^{C_{1}/2}\unicode[STIX]{x1D70C}$ . From these bounds, (4.20), and the local bilinearity of $\unicode[STIX]{x1D719}$ , we conclude (4.19) as desired.◻

Local $U^{2}$ -inverse theorem

The global inverse $U^{2}$ theorem, which is a simple and well-known exercise in discrete Fourier analysis, asserts that if a $1$ -bounded function $f:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{C}$ obeys the bound

(4.21) $$\begin{eqnarray}|\mathbb{E}f(\mathbf{h}_{0}+\mathbf{h}_{1})\overline{f}(\mathbf{h}_{0}+\mathbf{h}_{1}^{\prime })\overline{f}(\mathbf{h}_{0}^{\prime }+\mathbf{h}_{1})f(\mathbf{h}_{0}^{\prime }+\mathbf{h}_{1}^{\prime })|\geqslant \unicode[STIX]{x1D702}\end{eqnarray}$$

where $\mathbf{h}_{0},\mathbf{h}_{1},\mathbf{h}_{0}^{\prime },\mathbf{h}_{1}^{\prime }$ are drawn uniformly at random from $\mathbb{Z}/p\mathbb{Z}$ , then there exists $\unicode[STIX]{x1D709}\in \mathbb{Z}/p\mathbb{Z}$ such that

(4.22) $$\begin{eqnarray}|\mathbb{E}f(\mathbf{h})e_{p}(-\unicode[STIX]{x1D709}\mathbf{h})|\geqslant \unicode[STIX]{x1D702}^{1/2}\end{eqnarray}$$

where $\mathbf{h}$ is also drawn uniformly at random from $\mathbb{Z}/p\mathbb{Z}$ .

In this section we give a local version of the above claim, in which the random variables $\mathbf{h},\mathbf{h}_{0},\mathbf{h}_{1},\mathbf{h}_{0}^{\prime },\mathbf{h}_{1}^{\prime }$ are localized to a small Bohr set. If the rank of the Bohr set is bounded, one can modify the above arguments to obtain a reasonable inverse theorem of this nature, but in our application the rank of the Bohr set will be rather large, and it will be important that this rank does not affect the lower bound in correlations of the form (4.22). Fortunately, such a result is available, and will be crucial in the proofs of the two remaining claims (Corollary 4.13 and Theorem 8.1) needed to prove Theorem 1.1.

Here is a precise version of the claim.

Theorem 4.12. Let $S\subset \mathbb{Z}/p\mathbb{Z}$ be non-degenerate for some prime $p$ , and let $0<\unicode[STIX]{x1D702}<1/2$ . Let $\unicode[STIX]{x1D70C}_{0},\unicode[STIX]{x1D70C}_{1}$ be real parameters with $0<\unicode[STIX]{x1D70C}_{1}<\unicode[STIX]{x1D70C}_{0}<1/2$ and such that

(4.23) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{0}>\frac{C|S|}{\unicode[STIX]{x1D702}^{2}}\unicode[STIX]{x1D70C}_{1}\end{eqnarray}$$

for a sufficiently large absolute constant $C$ . Let $f:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{C}$ be a $1$ -bounded function such that

(4.24) $$\begin{eqnarray}|\mathbb{E}f(\mathbf{h}_{0}+\mathbf{h}_{1})\overline{f}(\mathbf{h}_{0}+\mathbf{h}_{1}^{\prime })\overline{f}(\mathbf{h}_{0}^{\prime }+\mathbf{h}_{1})f(\mathbf{h}_{0}^{\prime }+\mathbf{h}_{1}^{\prime })|\geqslant \unicode[STIX]{x1D702},\end{eqnarray}$$

where $\mathbf{h}_{0},\mathbf{h}_{0}^{\prime },\mathbf{h}_{1},\mathbf{h}_{1}^{\prime }$ are drawn independently and regularly from $B(S,\unicode[STIX]{x1D70C}_{0})$ , $B(S,\unicode[STIX]{x1D70C}_{0})$ , $B(S,\unicode[STIX]{x1D70C}_{1})$ , $B(S,\unicode[STIX]{x1D70C}_{1})$ respectively. Then there exists $\unicode[STIX]{x1D709}\in \mathbb{Z}/p\mathbb{Z}$ such that

$$\begin{eqnarray}\mathop{\sum }_{n_{0}\in \mathbb{Z}/p\mathbb{Z}}\mathbb{P}(\mathbf{n}_{0}=n_{0})|\mathbb{E}f(n_{0}+\mathbf{n}_{1})e_{p}(-\unicode[STIX]{x1D709}\mathbf{n}_{1})|^{2}\geqslant \unicode[STIX]{x1D702}/2\end{eqnarray}$$

where $\mathbf{n}_{0},\mathbf{n}_{1}$ are drawn independently and regularly from $B(S,\unicode[STIX]{x1D70C}_{0}),B(S,\unicode[STIX]{x1D70C}_{1})$ respectively.

Proof. We thank Fernando Shao for supplying a proof of this result, which was considerably simpler than our original argument.

For this proof, which is Fourier-analytic in nature, it will be convenient to work explicitly with probability densities rather than probabilistic notation. (However, in the lengthier proof of the local inverse $U^{3}$ theorem given in the next section, the probabilistic notation will be significantly cleaner to use.) In this argument, all sums will be over $\mathbb{Z}/p\mathbb{Z}$ . We abbreviate

$$\begin{eqnarray}\mathfrak{p}_{i}(h):=\mathfrak{p}_{B(S,\unicode[STIX]{x1D70C}_{i})}(h)=\mathbb{P}(\mathbf{h}_{i}=h)\end{eqnarray}$$

for $i=0,1$ and $h\in \mathbb{Z}/p\mathbb{Z}$ ; clearly we have $\mathfrak{p}_{i}(h)\geqslant 0$ and

(4.25) $$\begin{eqnarray}\mathop{\sum }_{h}\mathfrak{p}_{i}(h)=1.\end{eqnarray}$$

The hypothesis (4.24) may be written as

(4.26) $$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{h_{0},h_{0}^{\prime },h_{1},h_{1}^{\prime }}\mathfrak{p}_{0}(h_{0})\mathfrak{p}_{0}(h_{0}^{\prime })\mathfrak{p}_{1}(h_{1})\mathfrak{p}_{1}(h_{1}^{\prime })f(h_{0}+h_{1})\overline{f}(h_{0}+h_{1}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\overline{f}(h_{0}^{\prime }+h_{1})f(h_{0}^{\prime }+h_{1}^{\prime })\biggr|\geqslant \unicode[STIX]{x1D702}\end{eqnarray}$$

and our goal is to locate $\unicode[STIX]{x1D709}\in \mathbb{Z}/p\mathbb{Z}$ such that

$$\begin{eqnarray}\mathop{\sum }_{n_{0}}\mathfrak{p}_{0}(n_{0})\biggl|\mathop{\sum }_{n_{1}}\mathfrak{p}_{1}(n_{1})f(n_{0}+n_{1})e_{p}(-\unicode[STIX]{x1D709}n_{1})\biggr|^{2}\geqslant \unicode[STIX]{x1D702}/2.\end{eqnarray}$$

The first step is to replace the factor $\mathfrak{p}_{0}(h_{0})$ by the slightly different factor $\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1})\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1}^{\prime })$ . If we use the elementary inequality $|x^{1/2}-y^{1/2}|\leqslant |x-y|^{1/2}$ for $x,y\geqslant 0$ and then apply Cauchy–Schwarz, Lemma 4.4, and (4.23), we see that

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{h_{0}}|\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1})-\mathfrak{p}_{0}^{1/2}(h_{0})|\mathfrak{p}_{0}^{1/2}(h_{0})\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \mathop{\sum }_{h_{0}}|\mathfrak{p}_{0}(h_{0}+h_{1})-\mathfrak{p}_{0}(h_{0})|^{1/2}\mathfrak{p}_{0}^{1/2}(h_{0})\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \biggl(\mathop{\sum }_{h_{0}\in \mathbb{Z}/p\mathbb{Z}}|\mathfrak{p}_{0}(h_{0}+h_{1})-\mathfrak{p}_{0}(h_{0})|\biggr)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \quad =\biggl(\mathop{\sum }_{h_{0}\in \mathbb{Z}/p\mathbb{Z}}b_{h_{1}}(h_{0})\mathfrak{p}_{0}(h_{0}+h_{1})-b_{h_{1}}(h_{0})\mathfrak{p}_{0}(h_{0})\biggr)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \quad \ll \biggl(\frac{|S|\unicode[STIX]{x1D70C}_{1}}{\unicode[STIX]{x1D70C}_{0}}\biggr)^{1/2}\ll \frac{\unicode[STIX]{x1D702}}{C^{1/2}}\nonumber\end{eqnarray}$$

for any $h_{1}$ in the support of $\mathfrak{p}_{1}$ , where the $1$ -bounded function $b_{h_{1}}$ is given by $b_{h_{1}}(h_{0}):=\operatorname{sgn}(\mathfrak{p}_{0}(h_{0}+h_{1})-\mathfrak{p}_{0}(h_{0}))$ . Similarly we have

$$\begin{eqnarray}\mathop{\sum }_{h_{0}}|\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1}^{\prime })-\mathfrak{p}_{0}^{1/2}(h_{0})|\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1})\ll \frac{\unicode[STIX]{x1D702}}{C^{1/2}}\end{eqnarray}$$

whenever $h_{1}^{\prime }$ is also in the support of $\mathfrak{p}_{1}$ ; by the triangle inequality, we conclude that

$$\begin{eqnarray}\mathop{\sum }_{h_{0}}|\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1})\mathfrak{p}_{0}(h_{0}+h_{1}^{\prime })^{1/2}-\mathfrak{p}_{0}(h_{0})|\ll \frac{\unicode[STIX]{x1D702}}{C^{1/2}}\end{eqnarray}$$

for all $h_{1},h_{1}^{\prime }$ in the support of $\mathfrak{p}_{1}$ . From the $1$ -boundedness of $f$ and (4.25), we conclude that

$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{h_{0},h_{0}^{\prime },h_{1},h_{1}^{\prime }}|\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1})\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1}^{\prime })-\mathfrak{p}_{0}(h_{0})|\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathfrak{p}_{0}(h_{0}^{\prime })\mathfrak{p}_{1}(h_{1})\mathfrak{p}_{1}(h_{1}^{\prime })f(h_{0}+h_{1})\overline{f}(h_{0}+h_{1}^{\prime })\overline{f}(h_{0}^{\prime }+h_{1})f(h_{0}^{\prime }+h_{1}^{\prime })\biggr|\nonumber\\ \displaystyle & & \displaystyle \quad \ll \frac{\unicode[STIX]{x1D702}}{C^{1/2}}.\nonumber\end{eqnarray}$$

If $C$ is large enough, the left-hand side is thus bounded by $0.1\unicode[STIX]{x1D702}$ (for example), so by (4.26) and the triangle inequality we conclude that

$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{h_{0},h_{0}^{\prime },h_{1},h_{1}^{\prime }}\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1})\mathfrak{p}_{0}^{1/2}(h_{0}+h_{1}^{\prime })\mathfrak{p}_{0}(h_{0}^{\prime })\mathfrak{p}_{1}(h_{1})\mathfrak{p}_{1}(h_{1}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad \times \,f(h_{0}+h_{1})\overline{f}(h_{0}+h_{1}^{\prime })\overline{f}(h_{0}^{\prime }+h_{1})f(h_{0}^{\prime }+h_{1}^{\prime })\biggr|\geqslant 0.9\unicode[STIX]{x1D702}\nonumber\end{eqnarray}$$

If we write

(4.27) $$\begin{eqnarray}f_{0}(n):=f(n)\mathfrak{p}_{0}^{1/2}(n),\end{eqnarray}$$

we may rewrite the above estimate as

$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{h_{0},h_{0}^{\prime },h_{1},h_{1}^{\prime }}\mathfrak{p}_{0}(h_{0}^{\prime })\mathfrak{p}_{1}(h_{1})\mathfrak{p}_{1}(h_{1}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad \times \,f_{0}(h_{0}+h_{1})\overline{f_{0}}(h_{0}+h_{1}^{\prime })\overline{f}(h_{0}^{\prime }+h_{1})f(h_{0}^{\prime }+h_{1}^{\prime })\biggr|\geqslant 0.9\unicode[STIX]{x1D702}.\nonumber\end{eqnarray}$$

A similar argument then lets us replace $\mathfrak{p}_{0}(h_{0}^{\prime })$ with $\mathfrak{p}_{0}^{1/2}(h_{0}^{\prime }+h_{1})\mathfrak{p}_{0}^{1/2}(h_{0}^{\prime }+h_{1}^{\prime })$ , leaving us with

$$\begin{eqnarray}\displaystyle & & \displaystyle \biggl|\mathop{\sum }_{h_{0},h_{0}^{\prime },h_{1},h_{1}^{\prime }}\mathfrak{p}_{0}(h_{0}^{\prime }+h_{1})^{1/2}\mathfrak{p}_{0}(h_{0}^{\prime }+h_{1}^{\prime })^{1/2}\mathfrak{p}_{1}(h_{1})\mathfrak{p}_{1}(h_{1}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad \times \,f_{0}(h_{0}+h_{1})\overline{f_{0}}(h_{0}+h_{1}^{\prime })\overline{f}(h_{0}^{\prime }+h_{1})f(h_{0}^{\prime }+h_{1}^{\prime })\biggr|\geqslant 0.8\unicode[STIX]{x1D702}.\nonumber\end{eqnarray}$$

which we can simplify using (4.27) to

$$\begin{eqnarray}\biggl|\mathop{\sum }_{h_{0},h_{0}^{\prime },h_{1},h_{1}^{\prime }}\mathfrak{p}_{1}(h_{1})\mathfrak{p}_{1}(h_{1}^{\prime })f_{0}(h_{0}+h_{1})\overline{f_{0}}(h_{0}+h_{1}^{\prime })\overline{f_{0}}(h_{0}^{\prime }+h_{1})f_{0}(h_{0}^{\prime }+h_{1}^{\prime })\biggr|\geqslant 0.8\unicode[STIX]{x1D702}.\end{eqnarray}$$

Making the change of variables $n:=h_{1}-h_{1}^{\prime }$ , we may rewrite the left-hand side as

$$\begin{eqnarray}\mathop{\sum }_{n}(\mathfrak{p}_{1}\ast \tilde{\mathfrak{p}}_{1})(n)|(f_{0}\ast \tilde{f}_{0})(n)|^{2}\end{eqnarray}$$

where $\tilde{f}_{0}(n):=\overline{f_{0}}(-n)$ , and similarly for $p_{1}$ , and $f\ast g$ denotes the discrete convolution

$$\begin{eqnarray}f\ast g(n):=\mathop{\sum }_{m}f(m)g(n-m).\end{eqnarray}$$

Using the Fourier transform, we may then rewrite the previous bound as

(4.28) $$\begin{eqnarray}p^{4}\mathop{\sum }_{\unicode[STIX]{x1D709},\unicode[STIX]{x1D709}^{\prime }}|\hat{\mathfrak{p}}_{1}(\unicode[STIX]{x1D709}^{\prime })|^{2}|\hat{f}_{0}(\unicode[STIX]{x1D709})|^{2}|\hat{f}_{0}(\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}^{\prime })|^{2}\geqslant 0.8\unicode[STIX]{x1D702}\end{eqnarray}$$

where

$$\begin{eqnarray}\hat{f}(\unicode[STIX]{x1D709}):=\frac{1}{p}\mathop{\sum }_{n}f(n)e_{p}(-\unicode[STIX]{x1D709}n).\end{eqnarray}$$

From (4.25), the $1$ -boundedness of $f$ , and the Plancherel identity we have

$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D709}}|\hat{f}_{0}(\unicode[STIX]{x1D709})|^{2}=\frac{1}{p}\mathop{\sum }_{n}|f_{0}(n)|^{2}\leqslant \frac{1}{p}.\end{eqnarray}$$

By this, (4.28), and the pigeonhole principle, we may therefore find $\unicode[STIX]{x1D709}\in \mathbb{Z}/p\mathbb{Z}$ such that

$$\begin{eqnarray}p^{3}\mathop{\sum }_{\unicode[STIX]{x1D709}^{\prime }\in \mathbb{Z}/p\mathbb{Z}}|\hat{p}_{1}(\unicode[STIX]{x1D709}^{\prime })|^{2}|\hat{f}_{0}(\unicode[STIX]{x1D709}+\unicode[STIX]{x1D709}^{\prime })|^{2}\geqslant 0.8\unicode[STIX]{x1D702}.\end{eqnarray}$$

By the Plancherel identity again, the left-hand side may be rewritten as

$$\begin{eqnarray}\mathop{\sum }_{n_{0}}\biggl|\mathop{\sum }_{n_{1}}f_{0}(n_{0}-n_{1})\mathfrak{p}_{1}(n_{1})e_{p}(\unicode[STIX]{x1D709}n_{1})\biggr|^{2}\end{eqnarray}$$

and hence (by replacing $n_{1}$ with $-n_{1}$ and using (4.27))

$$\begin{eqnarray}\mathop{\sum }_{n_{0}}\biggl|\mathop{\sum }_{n_{1}}f(n_{0}+n_{1})\mathfrak{p}_{0}^{1/2}(n_{0}+n_{1})\mathfrak{p}_{1}(n_{1})e_{p}(-\unicode[STIX]{x1D709}n_{1})\biggr|^{2}\geqslant 0.8\unicode[STIX]{x1D702}.\end{eqnarray}$$

By argument similar to those at the beginning of the proof, we may replace $\mathfrak{p}_{0}^{1/2}(n_{0}+n_{1})$ by $\mathfrak{p}_{0}^{1/2}(n_{0})$ and conclude that

$$\begin{eqnarray}\mathop{\sum }_{n_{0}}\biggl|\mathop{\sum }_{n_{1}}f(n_{0}+n_{1})\mathfrak{p}_{0}^{1/2}(n_{0})\mathfrak{p}_{1}(n_{1})e(-\unicode[STIX]{x1D709}n_{1})\biggr|^{2}\geqslant 0.7\unicode[STIX]{x1D702},\end{eqnarray}$$

and the claim follows. ◻

As a corollary of this inverse theorem, we can establish that locally almost linear phases on Bohr sets can be approximated by globally linear phases; this will be needed in §7 to deal with poorly distributed quadratic factors.

Here is a precise statement.

Corollary 4.13. Let $\unicode[STIX]{x1D719}:n_{0}+B(S,\unicode[STIX]{x1D70C})\rightarrow \mathbb{R}/\mathbb{Z}$ be a function on a shifted Bohr set $n_{0}+B(S,\unicode[STIX]{x1D70C})$ that is “locally almost linear” in the sense that one has the bound

(4.29) $$\begin{eqnarray}\Vert \unicode[STIX]{x1D719}(n_{0}+h+k)-\unicode[STIX]{x1D719}(n_{0}+h)-\unicode[STIX]{x1D719}(n_{0}+k)+\unicode[STIX]{x1D719}(n_{0})\Vert _{\mathbb{R}/\mathbb{Z}}\leqslant A\frac{\Vert h\Vert _{S^{\bot }}\Vert k\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}^{2}}\quad\end{eqnarray}$$

for all $h,k\in B(S,\unicode[STIX]{x1D70C}/2)$ and some $A\geqslant 1$ . Then there exists $\unicode[STIX]{x1D709}\in \mathbb{Z}/p\mathbb{Z}$ such that

(4.30) $$\begin{eqnarray}\biggl\|\unicode[STIX]{x1D719}(n_{0}+h)-\unicode[STIX]{x1D719}(n_{0})-\frac{\unicode[STIX]{x1D709}h}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}\ll A^{1/2}|S|^{4}\frac{\Vert h\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}}\end{eqnarray}$$

for all $h\in B(S,\unicode[STIX]{x1D70C})$ .

Proof. By translating in space, we may normalize so that $n_{0}=0$ ; by shifting $\unicode[STIX]{x1D719}$ by a phase, we may also suppose that $\unicode[STIX]{x1D719}(0)=0$ . By replacing $\unicode[STIX]{x1D70C}$ with the smaller quantity $\unicode[STIX]{x1D70C}/A^{1/2}$ if necessary, we may normalize $A$ to be $1$ (note that (4.30) is trivial for $\Vert h\Vert _{S^{\bot }}\,\geqslant \,\unicode[STIX]{x1D70C}/A^{1/2}$ ). Thus, we now have a function $\unicode[STIX]{x1D719}\,:\,B(S,\unicode[STIX]{x1D70C})\rightarrow \mathbb{R}/\mathbb{Z}$ with $\unicode[STIX]{x1D719}(0)=0$ such that the quantity

(4.31) $$\begin{eqnarray}\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}(h,k):=\unicode[STIX]{x1D719}(h+k)-\unicode[STIX]{x1D719}(h)-\unicode[STIX]{x1D719}(k)\end{eqnarray}$$

obeys the bound

(4.32) $$\begin{eqnarray}\Vert \unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}(h,k)\Vert _{\mathbb{ R}/\mathbb{Z}}\leqslant \frac{\Vert h\Vert _{S^{\bot }}\Vert k\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}^{2}}\end{eqnarray}$$

for all $h,k\in B(S,\unicode[STIX]{x1D70C}/2)$ , and our task is to locate $\unicode[STIX]{x1D709}\in \mathbb{Z}/p\mathbb{Z}$ such that

(4.33) $$\begin{eqnarray}\biggl\|\unicode[STIX]{x1D719}(h)-\frac{\unicode[STIX]{x1D709}h}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}\ll |S|^{4}\frac{\Vert h\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}}\end{eqnarray}$$

for all $h\in B(S,\unicode[STIX]{x1D70C})$ .

Let $\unicode[STIX]{x1D70C}_{0}:=\unicode[STIX]{x1D70C}/100$ , and set $\unicode[STIX]{x1D70C}_{1}:=\unicode[STIX]{x1D70C}/C|S|^{3}$ for some sufficiently large absolute constant $C$ . If we let $f:\mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{C}$ be the $1$ -bounded function

(4.34) $$\begin{eqnarray}f(x):=1_{B(S,\unicode[STIX]{x1D70C})}e(\unicode[STIX]{x1D719}(x))\end{eqnarray}$$

and draw $\mathbf{h}_{0},\mathbf{h}_{0}^{\prime },\mathbf{h}_{1},\mathbf{h}_{1}^{\prime }$ independently and regularly from $B(S,\unicode[STIX]{x1D70C}_{0})$ , $B(S,\unicode[STIX]{x1D70C}_{0})$ , $B(S,\unicode[STIX]{x1D70C}_{1})$ , $B(S,\unicode[STIX]{x1D70C}_{1})$ respectively, then from (4.31) we have

$$\begin{eqnarray}\displaystyle & & \displaystyle f(\mathbf{h}_{0}+\mathbf{h}_{1})\overline{f}(\mathbf{h}_{0}+\mathbf{h}_{1}^{\prime })\overline{f}(\mathbf{h}_{0}^{\prime }+\mathbf{h}_{1})f(\mathbf{h}_{0}^{\prime }+\mathbf{h}_{1}^{\prime })\nonumber\\ \displaystyle & & \displaystyle \quad =e(\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}(\mathbf{h}_{0},\mathbf{h}_{1})-\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}(\mathbf{h}_{0}^{\prime },\mathbf{h}_{1})-\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}(\mathbf{h}_{0},\mathbf{h}_{1}^{\prime })+\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}(\mathbf{h}_{0}^{\prime },\mathbf{h}_{1}^{\prime })).\nonumber\end{eqnarray}$$

Applying (4.32) and taking expectations, we conclude that

$$\begin{eqnarray}|\mathbb{E}f(\mathbf{h}_{0}+\mathbf{h}_{1})\overline{f}(\mathbf{h}_{0}+\mathbf{h}_{1}^{\prime })\overline{f}(\mathbf{h}_{0}^{\prime }+\mathbf{h}_{1})f(\mathbf{h}_{0}^{\prime }+\mathbf{h}_{1}^{\prime })|\geqslant 1/2\end{eqnarray}$$

(for example). Applying Theorem 4.12 (which is applicable for $C$ large enough), we may thus find $\unicode[STIX]{x1D709}\in \mathbb{Z}/p\mathbb{Z}$ such that

$$\begin{eqnarray}\mathop{\sum }_{n_{0}\in \mathbb{Z}/p\mathbb{Z}}\mathbb{P}(\mathbf{n}_{0}=n_{0})|\mathbb{E}f(n_{0}+\mathbf{n}_{1})e_{p}(-\unicode[STIX]{x1D709}\mathbf{n}_{1})|^{2}\geqslant 1/4\end{eqnarray}$$

if $\mathbf{n}_{0},\mathbf{n}_{1}$ are drawn independently and regularly from $B(S,\unicode[STIX]{x1D70C}_{0}),B(S,\unicode[STIX]{x1D70C}_{1})$ respectively. In particular, there exists $n\in B(S,\unicode[STIX]{x1D70C}_{0})$ such that

$$\begin{eqnarray}|\mathbb{E}f(n+\mathbf{n}_{1})e_{p}(-\unicode[STIX]{x1D709}\mathbf{n}_{1})|\geqslant 1/4.\end{eqnarray}$$

By (4.34), (4.31) we have

$$\begin{eqnarray}f(n+\mathbf{n}_{1})=e(\unicode[STIX]{x1D719}(\mathbf{n}_{1})+\unicode[STIX]{x1D719}(n)+\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}(n,\mathbf{n}_{1}))\end{eqnarray}$$

so by (4.32) we conclude that

(4.35) $$\begin{eqnarray}\biggl|\mathbb{E}e\biggl(\unicode[STIX]{x1D719}(\mathbf{n}_{1})-\frac{\unicode[STIX]{x1D709}\mathbf{n}_{1}}{p}\biggr)\biggr|\gg 1.\end{eqnarray}$$

For any $h\in B(S,\unicode[STIX]{x1D70C}_{1})$ , we have from Lemma 4.4 that

$$\begin{eqnarray}\biggl|\mathbb{E}e\biggl(\unicode[STIX]{x1D719}(\mathbf{n}_{1}+h)-\frac{\unicode[STIX]{x1D709}(\mathbf{n}_{1}+h)}{p}\biggr)-\mathbb{E}e\biggl(\unicode[STIX]{x1D719}(\mathbf{n}_{1})-\frac{\unicode[STIX]{x1D709}\mathbf{n}_{1}}{p}\biggr)\biggr|\ll |S|\frac{\Vert h\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}_{1}};\end{eqnarray}$$

on the other hand, from (4.31) we have the identity

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathbb{E}e\biggl(\unicode[STIX]{x1D719}(\mathbf{n}_{1}+h)-\frac{\unicode[STIX]{x1D709}(\mathbf{n}_{1}+h)}{p}\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad =e\biggl(\unicode[STIX]{x1D719}(h)-\frac{\unicode[STIX]{x1D709}h}{p}\biggr)\mathbb{E}e\biggl(\unicode[STIX]{x1D719}(\mathbf{n}_{1})-\frac{\unicode[STIX]{x1D709}\mathbf{n}_{1}}{p}+\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}(\mathbf{n}_{1},h)\biggr).\nonumber\end{eqnarray}$$

Combining this with (4.32), (4.35), and (2.2), we conclude that

$$\begin{eqnarray}\biggl\|\unicode[STIX]{x1D719}(h)-\frac{\unicode[STIX]{x1D709}h}{p}\biggr\|_{\mathbb{R}/\mathbb{Z}}\asymp \biggl|e\biggl(\unicode[STIX]{x1D719}(h)-\frac{\unicode[STIX]{x1D709}h}{p}\biggr)-1\biggr|\ll |S|\frac{\Vert h\Vert _{S^{\bot }}}{\unicode[STIX]{x1D70C}_{1}}\end{eqnarray}$$

for all $h\in B(S,\unicode[STIX]{x1D70C}_{1})$ . As the claim (4.33) is trivial for $h\in B(S,\unicode[STIX]{x1D70C})\backslash B(S,\unicode[STIX]{x1D70C}_{1})$ , the claim follows.◻

5 Dilated tori

As mentioned in Example 3 of §3, to maintain good quantitative control (and specifically, Lipschitz norm control) on the functions $F:G\rightarrow [-1,1]$ used to build quadratic approximants, one needs to generalize the underlying domain $G$ to more general tori than the standard tori $(\mathbb{R}/\mathbb{Z})^{d}$ with the usual norm structure. It turns out that it will suffice to work with dilated tori of the form

$$\begin{eqnarray}G=\mathop{\prod }_{i=1}^{d}(\mathbb{R}/\unicode[STIX]{x1D706}_{i}\mathbb{Z}),\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{d}\geqslant 1$ are real numbers. One can view this dilated torus as the quotient of $\mathbb{R}^{d}$ by a dilated lattice $\unicode[STIX]{x1D6E4}:=\prod _{i=1}^{d}\unicode[STIX]{x1D706}_{i}\mathbb{Z}$ . We can place a “norm” on $G$ by declaring $\Vert x\Vert _{G}$ for $x\in G$ to be the Euclidean distance in $\mathbb{R}^{d}$ from $x$ to $\unicode[STIX]{x1D6E4}$ ; this generalizes the norm $\Vert \Vert _{\mathbb{R}/\mathbb{Z}}$ from §2. This in turn defines a metric $d_{G}$ on $G$ by the formula

$$\begin{eqnarray}d_{G}(x,y):=\Vert x-y\Vert _{G}.\end{eqnarray}$$

The volume $\operatorname{vol}(G)$ of a dilated torus is defined to be the product

$$\begin{eqnarray}\operatorname{vol}(G):=\mathop{\prod }_{i=1}^{d}\unicode[STIX]{x1D706}_{i}=\det (\unicode[STIX]{x1D6E4}).\end{eqnarray}$$

It will be important to keep this quantity under control during the iteration process. In particular, when transforming from one dilated torus to another, the volume of the new torus should behave like a linear function of the existing torus; anything worse than this (e.g. quadratic behaviour) will lead to undesirable bounds upon iteration.

We define the Pontryagin dual ${\hat{G}}$ of a dilated torus $G$ to be the lattice

$$\begin{eqnarray}{\hat{G}}:=\mathop{\prod }_{i=1}^{d}\frac{1}{\unicode[STIX]{x1D706}_{i}}\mathbb{Z}.\end{eqnarray}$$

Elements $k$ of this dual will be called dual frequencies of the torus. If $k=(k_{1},\ldots ,k_{d})$ is a dual frequency and $x=(x_{1},\ldots ,x_{d})$ is an element of $G$ , we define the dot product $k\cdot x\in \mathbb{R}/\mathbb{Z}$ in the usual fashion as

$$\begin{eqnarray}k\cdot x=k_{1}x_{1}+\cdots +k_{d}x_{d}\end{eqnarray}$$

noting that this gives a well-defined element of $\mathbb{R}/\mathbb{Z}$ .

A dual frequency $k$ is said to be irreducible if it is non-zero, and not of the form $k=nk^{\prime }$ for some other dual frequency $k^{\prime }$ and some natural number $n>1$ . If a dual frequency $k$ is irreducible, then its orthogonal complement

$$\begin{eqnarray}k^{\bot }:=\{x\in G:k\cdot x=0\}\end{eqnarray}$$

is a $(d-1)$ -dimensional subtorus of $G$ ; it inherits a metric $d_{k^{\bot }}$ from the torus $G$ it lies in. We will need to pass to such a complement when dealing with poorly distributed quadratic factors (as in the third or fourth examples in §3), however we encounter the technical issue that these complements $k^{\bot }$ will not quite be of the form of a dilated torus. However, we will be able to transform $k^{\bot }$ into a dilated torus using a bilipschitz transformation, as the following result shows.

Theorem 5.1. Let $G=\prod _{i=1}^{d}(\mathbb{R}/\unicode[STIX]{x1D706}_{i}\mathbb{Z})$ be a dilated torus, and let $k\in {\hat{G}}$ be an irreducible dual frequency of $G$ . Then there exists a dilated torus $G^{\prime }=\prod _{i=1}^{d-1}(\mathbb{R}/\unicode[STIX]{x1D706}_{i}^{\prime }\mathbb{Z})$ and a Lie group isomorphism $\unicode[STIX]{x1D713}:k^{\bot }\rightarrow G^{\prime }$ obeying the bilipschitz bounds

(5.1) $$\begin{eqnarray}\Vert \unicode[STIX]{x1D713}\Vert _{\operatorname{Lip}},\Vert \unicode[STIX]{x1D713}^{-1}\Vert _{\operatorname{ Lip}}\ll d^{O(d)}\end{eqnarray}$$

and such that one has the volume bound

(5.2) $$\begin{eqnarray}\operatorname{vol}(G^{\prime })=d^{O(d)}|k|\operatorname{vol}(G),\end{eqnarray}$$

where $|k|$ denotes the Euclidean magnitude of $k$ in $\mathbb{R}^{d}$ .

Proof. The case $d=0$ is vacuous and the case $d=1$ is trivial, so we may assume $d>1$ . One can identify $k^{\bot }$ with the quotient $V/\unicode[STIX]{x1D6E4}$ , where $V:=\{x\in \mathbb{R}^{d}:k\cdot x=0\}$ is the hyperplane in $\mathbb{R}^{d}$ orthogonal to $k$ (now viewed as an element of $\mathbb{R}^{d}$ ), and $\unicode[STIX]{x1D6E4}:=V\cap \prod _{i=1}^{d}(\unicode[STIX]{x1D706}_{i}\mathbb{Z})$ is the restriction of the lattice $\prod _{i=1}^{d}(\unicode[STIX]{x1D706}_{i}\mathbb{Z})$ to  $V$ .

As $k$ is irreducible, there exists a vector $e$ in the lattice $\prod _{i=1}^{d}(\unicode[STIX]{x1D706}_{i}\mathbb{Z})$ with $k\cdot e=1$ ; thus $e$ has distance $1/|k|$ to $V$ . One can form a fundamental domain of $\mathbb{R}^{d}/\prod _{i=1}^{d}(\unicode[STIX]{x1D706}_{i}\mathbb{Z})$ by taking any fundamental domain for $V/\unicode[STIX]{x1D6E4}$ and performing the Minkowski sum of that domain with the interval $\{te:0\leqslant t\leqslant 1\}$ . By Fubini’s theorem, the $d$ -dimensional Lebesgue measure of such a sum will equal the $(d-1)$ -dimensional Lebesgue measure of the fundamental domain of $V/\unicode[STIX]{x1D6E4}$ and $1/|k|$ ; thus the covolume of $\prod _{i=1}^{d}(\unicode[STIX]{x1D706}_{i}\mathbb{Z})$ in $\mathbb{R}^{d}$ equals $1/|k|$ times the covolume of $\unicode[STIX]{x1D6E4}$ in $V$ . As the former covolume (determinant) is $\prod _{i=1}^{d}\unicode[STIX]{x1D706}_{i}=\operatorname{vol}(G)$ , we conclude that $\unicode[STIX]{x1D6E4}$ has covolume $|k|\operatorname{vol}(G)$ in $V$ .

Applying Lemma 4.8, we can find linearly independent elements $v_{1},\ldots ,v_{d-1}$ generating $\unicode[STIX]{x1D6E4}$ such that

(5.3) $$\begin{eqnarray}B_{V}(0,O(d)^{-3d/2}t)\cap \unicode[STIX]{x1D6E4}\subset \biggl\{\mathop{\sum }_{i=1}^{r}n_{i}v_{i}:|n_{i}|\leqslant tN_{i}\biggr\}\subset B_{V}(0,t)\cap \unicode[STIX]{x1D6E4}\end{eqnarray}$$

for all $t>0$ , where $B_{V}(0,r)$ is the Euclidean ball of radius $r$ in $V$ , and the $n_{i}$ are understood to be integers, with the bound

(5.4) $$\begin{eqnarray}\mathop{\prod }_{i=1}^{d-1}N_{i}^{-1}=(2d)^{O(d)}|k|\operatorname{vol}(G).\end{eqnarray}$$

From (5.3) we conclude in particular that

(5.5) $$\begin{eqnarray}O(d)^{-3d/2}N_{i}^{-1}\leqslant |v_{i}|\leqslant N_{i}^{-1}\end{eqnarray}$$

for all $1\leqslant i\leqslant d$ .

We now define the $(d-1)$ -dimensional dilated torus

$$\begin{eqnarray}G^{\prime }:=\mathop{\prod }_{i=1}^{d-1}(\mathbb{R}/N_{i}^{-1}\mathbb{Z})\end{eqnarray}$$

and the isomorphism $\unicode[STIX]{x1D719}:V/\unicode[STIX]{x1D6E4}\rightarrow G^{\prime }$ by the formula

$$\begin{eqnarray}\unicode[STIX]{x1D719}\biggl(\mathop{\sum }_{i=1}^{d-1}t_{i}v_{i}~(\operatorname{mod}~\unicode[STIX]{x1D6E4})\biggr):=(t_{1}N_{1}^{-1},\ldots ,t_{d-1}N_{d-1}^{-1})\quad \biggl(\operatorname{mod}~\mathop{\prod }_{i=1}^{d-1}N_{i}^{-1}\mathbb{Z}\biggr)\end{eqnarray}$$

for real numbers $t_{1},\ldots ,t_{d-1}$ . It is easy to see that this is a Lie group isomorphism, and the bound (5.2) follows from (5.4). It remains to establish the bilipschitz bounds (5.1). It suffices to show that the linear isomorphism

$$\begin{eqnarray}\mathop{\sum }_{i=1}^{d-1}t_{i}v_{i}\mapsto (t_{1}N_{1}^{-1},\ldots ,t_{d-1}N_{d-1}^{-1})\end{eqnarray}$$

from $V$ to $\mathbb{R}^{d-1}$ , together with its inverse, have an operator norm of $O(d^{O(d)})$ . For the inverse map, this is clear from (5.5). For the forward map, it suffices from Cramer’s rule to show that

$$\begin{eqnarray}\frac{|v_{1}\wedge \cdots \wedge v_{i-1}\wedge x\wedge v_{i+1}\wedge \cdots \wedge v_{d-1}|}{|v_{1}\wedge \cdots \wedge v_{d-1}|}\ll \frac{d^{O(d)}}{\unicode[STIX]{x1D706}_{i}^{\prime }}\end{eqnarray}$$

for all $i=1,\ldots ,d-1$ and all unit vectors $x$ in $V$ . But from (5.5) the numerator is at most $\prod _{1\leqslant i^{\prime }\leqslant d-1:i^{\prime }\neq i}N_{i^{\prime }}^{-1}$ , while the denominator is the volume of a fundamental domain in $V$ and is thus equal to $d^{O(d)}N_{1}^{-1}\ldots N_{d-1}^{-1}$ thanks to (5.4). The claim follows.◻

6 Constructing the approximants

In this section we construct the abstract directed graph $G=(V,E)$ that appears in Proposition 3.3. For the rest of the paper, the prime $p$ , the function $f:\mathbb{Z}/p\mathbb{Z}\rightarrow [-1,1]$ , and the parameter $\unicode[STIX]{x1D702}$ with $0<\unicode[STIX]{x1D702}\leqslant {\textstyle \frac{1}{10}}$ are fixed, and we assume that (3.21) holds.

We begin with a description of the structured approximants $v\in V$ .

Definition 6.1 (Structured local approximant).

A structured local approximant is a tuple

$$\begin{eqnarray}v=(C,\mathbf{c},(n_{c}+B(S_{c},\unicode[STIX]{x1D70C}_{c}))_{c\in C},(G_{c})_{c\in C},(F_{c})_{c\in C},(\unicode[STIX]{x1D6EF}_{c})_{c\in C})\end{eqnarray}$$

consisting of the following objects:

• ∙ a finite non-empty set $C$ ;

• ∙ a random variable $\mathbf{c}$ , which we call the label variable, taking values in $C$ ;

• ∙ a shifted Bohr set $n_{c}+B(S_{c},\unicode[STIX]{x1D70C}_{c})$ associated to each label $c\in C$ ;

• ∙ a dilated torus $G_{c}$ associated to each label $c\in C$ ;

• ∙ a $1$ -Lipschitz function $F_{c}:G_{c}\rightarrow [-1,1]$ associated to each label $c\in \mathbb{C}$ ; and

• ∙ a locally quadratic function $\unicode[STIX]{x1D6EF}_{c}:n_{c}+B(S_{c},\unicode[STIX]{x1D70C}_{c})\rightarrow G_{c}$ associated to each label $c\in C$ .

We denote the collection of all structured local approximants (up to isomorphismFootnote ³ ) as $V$ . Given any structured local approximant $v\in V$ , we define the random variables $(\mathbf{a}_{v},\mathbf{r}_{v},\mathbf{f}_{v})$ associated to $v$ by the following construction.

1. (1) First, let $\mathbf{c}$ be the random label variable appearing above.

2. (2) For each $c\in C$ in the essential range of $\mathbf{c}$ , if we condition on the event $\mathbf{c}=c$ , we draw $\mathbf{a}_{v},\mathbf{r}_{v}$ independently and regularly from $n_{c}+B(S_{c},\unicode[STIX]{x1D70C}_{c}/2)$ and $B(S_{c},\exp (-\unicode[STIX]{x1D702}^{-C_{4}})\unicode[STIX]{x1D70C}_{c})$ respectively, and then we let $\mathbf{f}_{v}$ be the function

   $$\begin{eqnarray}\mathbf{f}_{v}(a):=F_{c}(\unicode[STIX]{x1D6EF}_{c}(a)).\end{eqnarray}$$

Thus $\mathbf{f}_{v}$ is deterministic when $\mathbf{c}$ is conditioned to be fixed, but random when $\mathbf{c}$ is allowed to vary.

We also define the following additional statistics of the structured local approximant $v$ :

• ∙ the waste $\operatorname{waste}(v)$ is the quantity $|\mathbb{E}f(\mathbf{a})-\mathbb{E}_{a\in \mathbb{Z}/p\mathbb{Z}}f(a)|$ ;

• ∙ the $1$ -error $\operatorname{Err}_{1}(v)$ is $|\mathbb{E}\mathbf{f}(\mathbf{a})-\mathbb{E}f(\mathbf{a})|$ ;

• ∙ the $4$ -error $\operatorname{Err}_{4}(v)$ is $|\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(\mathbf{f})-\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f)|$ ;

• ∙ the energy $\operatorname{Energy}(v)$ is $\mathbb{E}|f(\mathbf{a})-\mathbf{f}(\mathbf{a})|^{2}$ ;

• ∙ the linear rank $d_{1}(v)$ is $\max _{c\in C}|S_{c}|$ ;

• ∙ the quadratic dimension $d_{2}(v)$ is $\max _{c\in C}\dim (G_{c})$ ;

• ∙ the linear scale $\unicode[STIX]{x1D70C}(v)$ is $\min _{c\in C}\unicode[STIX]{x1D70C}_{c}$ ;

• ∙ the quadratic volume $\operatorname{vol}(v)$ is the quantity $\max _{c\in C}\operatorname{vol}(G_{c})$ ;

• ∙ the poorly distributed quadratic dimension $d_{2}^{\text{poor}}(v)$ is the maximum value of $\dim (G_{c})$ over all poorly distributed $c$ in the essential range of $\mathbf{c}$ , or zero if no such $c$ exists. Here, an element $c$ in the essential range of $\mathbf{c}$ is said to be poorly distributed if one has

  (6.1) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{\mathbf{a},\mathbf{r}}(f|\mathbf{c}=c)<\mathbb{E}(\mathbf{f}(\mathbf{a})|\mathbf{c}=c)^{4}-\frac{\unicode[STIX]{x1D702}}{2}.\end{eqnarray}$$

This gives the set $V$ of structured local approximants for Proposition 3.3; we clearly have $0\leqslant d_{2}^{\text{poor}}(v)\leqslant d_{2}(v)$ for all $v\in V$ .

We now also define the initial approximant.

Definition 6.2. The initial approximant $v_{0}\in V$ is defined to be the tuple

$$\begin{eqnarray}v_{0}=(C,\mathbf{c},(n_{c}+B(S_{c},\unicode[STIX]{x1D70C}_{c}))_{c\in C},(G_{c})_{c\in C},(F_{c})_{c\in C},(\unicode[STIX]{x1D6EF}_{c})_{c\in C})\end{eqnarray}$$

defined as follows:

• ∙ $C:=\mathbb{Z}/p\mathbb{Z}$ , and $\mathbf{c}$ is drawn uniformly from $C$ ;

• ∙ for each $c\in C$ , we have $n_{c}:=0$ , $S_{c}:=\{1\}$ , and $\unicode[STIX]{x1D70C}_{c}:=1$ ;

• ∙ for each $c\in C$ , the group $G_{c}$ is the standard $0$ -torus $(\mathbb{R}/\mathbb{Z})^{0}$ (that is to say, a point);

• ∙ for each $c\in C$ , the function $F_{c}:G_{c}\rightarrow [-1,1]$ is the zero function $F_{c}(x):=0$ ;

• ∙ for each $c\in C$ , the function $\unicode[STIX]{x1D6EF}_{c}:\mathbb{Z}/p\mathbb{Z}\rightarrow G_{c}$ is the unique (constant) map from $\mathbb{Z}/p\mathbb{Z}$ to the point $G_{c}$ .

By chasing the definitions, we see that $\mathbf{a}_{v_{0}}$ is uniformly distributed in $\mathbb{Z}/p\mathbb{Z}$ , and we can compute several of the statistics of the initial approximant $v_{0}$ :

(6.2) $$\begin{eqnarray}\operatorname{waste}(v_{0})=d_{2}^{\text{poor}}(v_{0})=d_{2}(v_{0})=0;d_{1}(v_{0})=\unicode[STIX]{x1D70C}(v)=\operatorname{vol}(v)=1.\end{eqnarray}$$

Now we define the edges of the graph $G(V,E)$ .

Definition 6.3. We let $E$ be the set of all directed edges $v\rightarrow v^{\prime }$ , where $v,v^{\prime }\in V$ are structured local approximants such that

$$\begin{eqnarray}\displaystyle d_{1}(v^{\prime }) & {\leqslant} & \displaystyle d_{1}(v)+\unicode[STIX]{x1D702}^{-C_{2}},\nonumber\\ \displaystyle d_{2}(v^{\prime }) & {\leqslant} & \displaystyle d_{2}(v)+1,\nonumber\\ \displaystyle \unicode[STIX]{x1D70C}(v^{\prime }) & {\geqslant} & \displaystyle \exp (-\unicode[STIX]{x1D702}^{-C_{5}})\unicode[STIX]{x1D70C}(v),\nonumber\\ \displaystyle \operatorname{vol}(v^{\prime }) & {\leqslant} & \displaystyle \exp (\unicode[STIX]{x1D702}^{-C_{3}})\operatorname{vol}(v),\nonumber\\ \displaystyle |\text{waste}(v)-\operatorname{waste}(v^{\prime })| & {\leqslant} & \displaystyle \unicode[STIX]{x1D702}^{C_{3}}.\nonumber\end{eqnarray}$$

From this definition and (6.2) we have the following bounds on the various statistics of vertices of $V$ that are not too far from the initial vertex $v_{0}$ , assuming that each constant $C_{i}$ is chosen sufficiently large depending on the preceding constants $C_{1},\ldots ,C_{i-1}$ .

Lemma 6.4. Suppose a vertex $v=v_{k}\in V$ can be reached from $v_{0}$ by a path $v_{0}\rightarrow v_{1}\rightarrow \cdots \rightarrow v_{k}$ with $0\leqslant k\leqslant 8\unicode[STIX]{x1D702}^{-2C_{2}}$ . Then we have

(6.3) $$\begin{eqnarray}\displaystyle d_{1}(v) & {\leqslant} & \displaystyle 8\unicode[STIX]{x1D702}^{-3C_{2}},\end{eqnarray}$$

(6.4) $$\begin{eqnarray}\displaystyle d_{2}(v) & {\leqslant} & \displaystyle 8\unicode[STIX]{x1D702}^{-2C_{2}},\end{eqnarray}$$

(6.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}(v) & {\geqslant} & \displaystyle \exp (-\unicode[STIX]{x1D702}^{-2C_{5}}),\end{eqnarray}$$

(6.6) $$\begin{eqnarray}\displaystyle \operatorname{vol}(v) & {\leqslant} & \displaystyle \exp (\unicode[STIX]{x1D702}^{-2C_{3}}),\end{eqnarray}$$

(6.7) $$\begin{eqnarray}\displaystyle \operatorname{waste}(v) & {\leqslant} & \displaystyle \unicode[STIX]{x1D702}^{C_{3}/2}.\end{eqnarray}$$