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 4 (A poorly distributed impure quadratic factor).

We now consider an example that is in some sense a combination of the second and third examples. Namely, we suppose we are in the same situation as in the second example, except that we allow some of the indices $c$ to have “poor quadratic distribution” in the sense that they admit non-trivial relations of the form (3.10). Again we may assume without loss of generality that $k_{d_{c}}$ is non-zero in such relations. Because of such relations, we no longer expect to have the equidistribution properties that were used in the second example. However, by modifying the calculations in the third example, we can obtain a new representation of $f$ (again allowing for a small error) on each of the progressions $P_{c}$ with poor quadratic distribution to reduce the number $d_{c}$ of quadratic polynomials used in that progression by one. Iterating this process a finite number of times, one eventually returns to the situation in the second example in which no non-trivial relations occur, at which point one can (heuristically, at least) verify Theorem 3.1 in this case.

The situation becomes slightly more complicated if one adds a lower order linear term $\unicode[STIX]{x1D701}_{c,i}n$ to the purely quadratic phases $\unicode[STIX]{x1D709}_{c,i}n^{2}$ appearing in the second example; this basically is the type of situation one encounters for instance at the conclusion of the third example. In this case, every time one converts a non-trivial relation of the form (3.10) on one of the cells $P_{c}$ of the partition into a new representation of $f$ on that cell, one must subdivide that cell $P_{j}$ into smaller pieces, by intersecting $P_{j}$ with various Bohr sets. However, the resulting sets still behave somewhat like arithmetic progressions, and it turns out that we can still iterate the construction a bounded number of times until no further non-trivial relations between surviving quadratic phases remain on any of the cells of the partition, at which point one can (heuristically, at least) verify Theorem 3.1 in this case (as well as in the case considered in the third 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.