2.1 General notation

Our notation largely follows [Reference Krause, Mirek and Tao13], though somewhat abridged, as some of the notation in [Reference Krause, Mirek and Tao13] is only used to establish results or arguments that we are treating here as ‘black boxes’.

We use to denote the positive integers and to denote the natural numbers.

We use to denote the indicator function of a set E. Similarly, if S is a statement, we use to denote its indicator, equal to $1$ if S is true and $0$ if S is false. Thus, for instance, . We use $|E|$ to denote the cardinality of a set E and adopt for $f\colon E\to \mathbb {C}$ the averaging notation

if E is finite and non-empty. We similarly define $L^p$ norms

for $0 < p < \infty $ , with the usual convention that $\|f\|_{L^\infty (E)}$ is the (essential) supremum of f on E. One can extend these averaging conventions to other measurable spaces E of positive finite measure (such as a p-adic group $\mathbb {Z}_p$ equipped with Haar probability measure), if f (or $|f|^p$ ) is absolutely integrable, in the obvious fashion. When X is equipped with counting measure, we will write $\ell ^p(X)$ or just $\ell ^{p}$ in place of $L^p(X)$ .

Throughout, $p'$ denotes the dual exponent of $p\in [1,\infty ]$ , so $1/p+1/p'=1$ .

If $f \colon X \to \mathbb {C}$ , $g \colon Y \to \mathbb {C}$ are functions, we use $f \otimes g \colon X \times Y \to \mathbb {C}$ to denote the tensor product

2.2 Magnitudes and asymptotic notation

We use the Japanese bracket notation

for any real or complex x. We use $\lfloor x \rfloor $ to denote the greatest integer less than or equal to x. For any $N \geq 1$ , we define the logarithmic scale $\operatorname {Log} N$ of N by the formula

(2.1)

thus $\operatorname {Log} N$ is the unique natural number such that $2^{\operatorname {Log} N} \leq N < 2^{\operatorname {Log} N+1}$ .

For any two quantities $A, B$ , we will write $A \lesssim B$ , $B \gtrsim A$ or $A = O(B)$ to denote the bound $|A| \leq CB$ for some absolute constant C. If we need the implied constant C to depend on additional parameters, we will denote this by subscripts; thus, for instance, $A \lesssim _\rho B$ denotes the bound $|A| \leq C_\rho B$ for some $C_\rho $ depending on $\rho $ . We write $A \sim B$ for $A \lesssim B \lesssim A$ . To abbreviate the notation, we will sometimes explicitly permit the implied constant to depend on certain fixed parameters (such as the polynomial P) when the issue of uniformity with respect to such parameters is not of relevance. Due to our reliance in some places on tools based on Siegel’s theorem (specifically, Siegel’s theorem is used in [Reference Matomäki, Shao, Tao and Teräväinen18], and we will use results from that paper to establish equation (3.1)), several of the implied constants in our arguments will be ineffective, but we will not track the effectivity of constants explicitly in this paper.

2.3 Algebraic notation

If R is a commutative ring, we use $R^\times $ to denote the multiplicatively invertible elements of R.

2.4 Number theoretic notation

For any $N> 0$ , $[N]$ denotes the discrete interval . If $q_1,q_2 \in \mathbb {Z}_+$ , we write $q_1\mid q_2$ if $q_1$ divides $q_2$ . If $a,q \in \mathbb {Z}_+$ , we let $(a,q)$ denote the greatest common divisor of a and q, and $[a,q]$ the least common multiple.

All sums and products over the symbol p will be understood to be over primes; other sums will be understood to be over positive integers unless otherwise specified.

In addition to the von Mangoldt function $\Lambda (n)$ and Möbius function $\mu (n)$ already introduced, we will also use the divisor function and the Euler totient function .

2.5 Fourier analytic notation

We write for any real $\theta $ , and also $\|\theta \|_{\mathbb {R}/\mathbb {Z}}$ for the distance from $\theta $ to the nearest integer.

For a prime p, we let $\mathbb {Z}_p$ be the ring of p-adic integers, defined as the inverse limit of the cyclic groups $\mathbb {Z}/p^j\mathbb {Z}$ for $j \in \mathbb {N}$ ; this is a compact abelian group equipped with a Haar probability measure. Similarly, let $\hat {\mathbb {Z}}$ be the ring of profinite integers, defined as the inverse limit of the cyclic groups $\mathbb {Z}/Q\mathbb {Z}$ for all positive integers Q; this is again a compact abelian group with a Haar probability measure, being the direct product of the $\mathbb {Z}_p$ . We use $\mathbb {E}_{\mathbb {Z}_p}$ or $\mathbb {E}_{\hat {\mathbb {Z}}}$ to denote averaging with respect to these compact abelian groups. Finally, we let denote the ring of adelic integers, which is a locally compact abelian group.

We define some Fourier transforms on various locally compact abelian groups.

1. (i) Given a summable function $f \colon \mathbb {Z} \to \mathbb {C}$ , the Fourier transform $\mathcal {F}_{\mathbb {Z}} f \colon \mathbb {R}/\mathbb {Z} \to \mathbb {C}$ is defined by the formula

   

2. (ii) Given a Schwartz function $f \colon \mathbb {R} \to \mathbb {C}$ , the Fourier transform $\mathcal {F}_{\mathbb {R}} f \colon \mathbb {R} \to \mathbb {C}$ is defined by the formula

   

3. (iii) Given a function $f \colon \hat {\mathbb {Z}} \to \mathbb {C}$ which is Schwartz–Bruhat in the sense that it factors through a function $f_Q \colon \mathbb {Z}/Q\mathbb {Z} \to \mathbb {C}$ on a cyclic group, we define the Fourier transform $\mathcal {F}_{\hat {\mathbb {Z}}} f \colon \mathbb {Q}/\mathbb {Z} \to \mathbb {C}$ by the formula

   
   for any integer a.

4. (iv) Given a function $f \colon \mathbb {A}_{\mathbb {Z}} \to \mathbb {C}$ which is Schwartz–Bruhat in the sense that it factors through a function $f_Q \colon \mathbb {R} \times \mathbb {Z}/Q\mathbb {Z}$ which is Schwartz in the first variable, we define the Fourier transform $\mathcal {F}_{\mathbb {A}} f \colon \mathbb {R} \times \mathbb {Q}/\mathbb {Z} \to \mathbb {C}$ by the formula

   
   for integer a and $\xi \in \mathbb {R}$ , and $\mathcal {F}_{\hat {\mathbb {A}}}$ vanishing otherwise.

We refer the reader to [Reference Krause, Mirek and Tao13, §4] for a further discussion of the Fourier transform on such locally compact abelian groups as $\mathbb {Z}$ , $\mathbb {R}$ , $\mathbb {Z}_p$ , $\hat {\mathbb {Z}}$ , $\mathbb {Z}/Q\mathbb {Z}$ or $\mathbb {A}_{\mathbb {Z}}$ , and the various intertwining relationships among these transforms.

Given a Schwartz symbol $m \colon \mathbb {R}/\mathbb {Z} \to \mathbb {C}$ , we define the Fourier multiplier $\mathrm {T}_m$ on $\ell ^2(\mathbb {Z})$ by the formula

and, similarly, given a bilinear Schwartz symbol $m \colon \mathbb {R}/\mathbb {Z} \times \mathbb {R}/\mathbb {Z} \to \mathbb {C}$ , define the bilinear Fourier multiplier $\mathrm {B}_m$ by the formula

Linear and bilinear multipliers are defined similarly for the other locally compact abelian groups defined here, and obey a certain operator calculus; again, we refer the reader to [Reference Krause, Mirek and Tao13, §4] for details, as we shall largely use facts and arguments about these operators from [Reference Krause, Mirek and Tao13] as ‘black boxes’.

We will need the Ionescu–Wainger Fourier multipliers on major arcs. Again, we shall mostly be using these tools as ‘black boxes’, so their definition and properties are not of critical importance in this paper; however, for sake of completeness, we recall the main definitions from [Reference Krause, Mirek and Tao13]. Given a small parameter $\rho $ , it is possible to assign a Ionescu–Wainger height $\mathrm {h}(\alpha )=\mathrm {h}_{\rho }(\alpha ) \in 2^{\mathbb {N}}$ for each $\alpha \in \mathbb {Q}/\mathbb {Z}$ ; see [Reference Krause, Mirek and Tao13, Appendix A]. Using this height, we define the Ionescu–Wainger arithmetic frequency sets

and the Ionescu–Wainger major arcs

(2.2)

thus, ${\mathcal M}_{\leq l, \leq k}$ is the union of arcs $[\alpha -2^k, \alpha +2^k]$ for $\alpha \in (\mathbb {Q}/\mathbb {Z})_{\leq l}$ ; we will be focused on the regime where k is sufficiently small that these arcs are disjoint, which happens whenever $k \leq -C_\rho 2^{\rho l}$ . We also use the variants

and

with the convention that $(\mathbb {Q}/\mathbb {Z})_{\leq -1}$ and ${\mathcal M}_{\leq -1,k}$ are empty.

The Ionescu–Wainger Fourier projection operator $\Pi _{\leq l, \leq k}$ for any $(l,k) \in \mathbb {N} \times \mathbb {Z}$ is defined by the formula

$$ \begin{align*} \Pi_{\leq l, \leq k} f(x) = \sum_{\alpha \in (\mathbb{Q}/\mathbb{Z})_{\leq l}} \int_{\mathbb{R}} \eta(\theta/2^k) \mathcal{F}_{\mathbb{Z}} f(\alpha+\theta) e(-x(\alpha+\theta))\, d\theta,\end{align*} $$

where $\eta $ is a smooth even function supported on $[-1,1]$ that equals $1$ on $[-1/2,1/2]$ . We then define

We refer the reader to [Reference Krause, Mirek and Tao13, §5, Appendix A] for the key properties of these projections, which can be viewed as analogues of Littlewood–Paley projection operators for major arcs.

2.6 Variational norms

A sequence $1 \leq N_1 < \cdots < N_k$ of positive reals is said to be $\unicode{x3bb} $ -lacunary for some $\unicode{x3bb} \geq 1$ if

$$ \begin{align*} N_{j+1}/N_j> \unicode{x3bb}\end{align*} $$

for all $1 \leq j < k$ .

For any finite dimensional normed vector space $(B,\|\cdot \|_B)$ and any sequence $(\mathfrak a_t)_{t\in \mathbb {I}}$ of elements of B indexed by a totally ordered set $\mathbb {I}$ , and any exponent $1 \leq r < \infty $ , the r-variation seminorm is defined by the formula

(2.3)

where the supremum is taken over all finite increasing sequences in $\mathbb {I}$ and is set by convention to equal zero if $\mathbb {I}$ is empty.

The r-variation norm for $1 \leq r < \infty $ is defined by

(2.4)

This clearly defines a norm on the space of functions from $\mathbb {I}$ to B. If $B=\mathbb {C}$ , then we will abbreviate $V^r(\mathbb {I};X)$ to $V^r(\mathbb {I})$ or $V^r$ , and $\mathbf {V}^r(\mathbb {I};X)$ to $ \mathbf {V}^r(\mathbb {I})$ or $\mathbf {V}^r$ .

2.7 Gowers norms

In addition to the little Gowers uniformity norm $u^{d+1}[N]$ defined in equation (1.8), we will also need the full Gowers norm $U^{d+1}[N]$ defined for functions $f \colon \mathbb {Z} \to \mathbb {C}$ as

where the $U^{d+1}(\mathbb {Z})$ norm is defined for finitely supported functions by the formula

where $\omega = (\omega _1,\ldots ,\omega _{d+1})$ and ${\mathcal C}$ denotes the complex conjugation operator. It is well known that

(2.5) $$ \begin{align} \| f\|_{u^{d+1}[N]} \lesssim_d \|f\|_{U^{d+1}[N]}; \end{align} $$

see, e.g. [Reference Green and Tao5, equation (2.2)].

Similar uniformity norms $u^{d+1}(I)$ , $U^{d+1}(I)$ can then be defined for other intervals $I \subset \mathbb {R}$ than $[N]$ in the obvious fashion.

4.1 Bounds on the Cramér approximant

We begin with the Cramér approximant. First, we record an easy uniform bound.

Lemma 4.1. (Uniform bound on Cramér model)

If $w \geq 1$ , then

$$ \begin{align*} 0 \leq \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w}(n) \lesssim \langle \operatorname{Log} w\rangle \end{align*} $$

for all $n \in \mathbb {Z}$ .

Proof. This is immediate from the Mertens theorem bound

$$ \begin{align*}\frac{W}{\varphi(W)} = \prod_{p \leq w} \frac{p}{p-1} \lesssim \langle \operatorname{Log} w\rangle.\\[-46pt]\end{align*} $$

The Cramér approximant is not easily expressible as an exact Type I sum once w is reasonably large (in particular, larger than $\operatorname {Log} N$ ), but thanks to the fundamental lemma of sieve theory, it can be approximated by such a sum.

Lemma 4.2. (Fundamental lemma of sieve theory)

If $2 \leq w \leq y \leq N^{1/10}$ , then there exist weights $\unicode{x3bb} ^\pm _{\ell } \in [-1,1]$ , supported on $1\leq \ell \leq y$ , such that

$$ \begin{align*} \sum_{\ell\mid n} \unicode{x3bb}^-_{\ell} \leq \frac{\varphi(W)}{W} \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w}(n) \leq \sum_{\ell\mid n} \unicode{x3bb}^+_{\ell}\end{align*} $$

for all n, and also

$$ \begin{align*} \mathbb{E}_{n \in I} \sum_{\ell\mid n} \unicode{x3bb}^\pm_{\ell} = \frac{\varphi(W)}{W} (1 + O( \exp( -\!\log y/ \log w ))) \end{align*} $$

for any interval I of length N. In particular,

$$ \begin{align*} \mathbb{E}_{n \in I} \bigg|\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w}(n) - \frac{W}{\varphi(W)} \sum_{\ell\mid n} \unicode{x3bb}^\pm_{\ell}\bigg| \lesssim \exp(-\!\log y /\log w ).\end{align*} $$

The fundamental lemma can then be used to give many good estimates for the Cramér model.

Proposition 4.3. (Linear equations in the Cramér model)

Let $t,m \geq 1$ be integers and let $N \geq 100$ . Let $\Omega \subset [-N,N]^d$ be convex, and let $\psi _1,\ldots ,\psi _t \colon \mathbb {Z}^m \to \mathbb {Z}$ be linear forms

$$ \begin{align*} \psi_i(\vec n) = \vec n \cdot \dot \psi_i + \psi_i(0) \end{align*} $$

for some $\dot \psi _i \in \mathbb {Z}^m$ and $\psi _i(0) \in \mathbb {Z}$ . Assume that the linear coefficients $\dot \psi _1,\ldots ,\dot \psi _t \in \mathbb {Z}^m$ are all pairwise linearly independent and have magnitude at most $\exp (\log ^{3/5} N)$ . Suppose that $1 \leq z_i \leq \exp (\operatorname {Log}^{1/10} N)$ for all $i=1,\ldots ,t$ . Then, one has

$$ \begin{align*} \sum_{\vec n \in \Omega \cap \mathbb{Z}^m} \prod_{i=1}^t \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r},z_i}(\psi_i(\vec n)) = \mathrm{vol}(\Omega) \prod_p \beta_p + O_{t,m}( N^m \exp(-c \operatorname{Log}^{4/5} N))\end{align*} $$

for some $c>0$ depending only on $t,m$ , where for each p, $\beta _p$ is the local factor

where $\psi _i$ is also viewed as a map from $(\mathbb {Z}/p\mathbb {Z})^m$ to $\mathbb {Z}/p\mathbb {Z}$ in the obvious fashion. Furthermore, $\beta _p$ obeys the bounds

(4.4) $$ \begin{align} \beta_p = 1 + O_{t,m}(1/p^2) \end{align} $$

for all primes p (and $\beta _p=1$ if $p> \max (z_1,\ldots ,z_t)$ ).

Specializing to the $t=m=1$ case (and noting that the constant coefficients of $\psi _i$ can be large in Proposition 4.3), we immediately obtain the following corollary.

Corollary 4.4. (Mean value of Cramér)

Let $N \geq 100$ and $1 \leq z \leq \exp (\operatorname {Log}^{1/10} N)$ , then

$$ \begin{align*} \mathbb{E}_{n \in I} \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, z}(n) = 1 + O(\exp(-c \operatorname{Log}^{4/5} N))\end{align*} $$

for any interval I of length N. In particular, since $\Lambda _{\mathrm{Cram}\acute{\rm e}\mathrm{r},z}(n)$ is non-negative, we also have

$$ \begin{align*} \mathbb{E}_{n \in I} |\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, z}(n)| = 1 + O(\exp(-c \operatorname{Log}^{4/5} N)).\end{align*} $$

More generally, if $1 \leq q \leq z$ and $a\ (q)$ is a residue class, then

As a more sophisticated application of Proposition 4.3, we record the following improvement of [Reference Tao and Teräväinen25, Proposition 1.2].

Lemma 4.5. (Improved stability of the Cramér model)

If $1 \leq z,w \leq \exp (\operatorname {Log}^{1/10} N)$ , for any $d \ge 1$ , one has

$$ \begin{align*} \| \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w} - \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, z} \|_{U^{d+1}(I)} \lesssim_{d} w^{-c} + z^{-c}\end{align*} $$

for any interval I of length N. In particular, by equation (2.5),

$$ \begin{align*} \| \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w} - \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, z} \|_{u^{d+1}(I)} \lesssim_{d} w^{-c} + z^{-c}.\end{align*} $$

In fact, one can take $c = 1/2^{d+1}$ in these estimates.

The result in [Reference Tao and Teräväinen25, Proposition 1.2] had an additional term of $\operatorname {Log}^{-c} N$ on the right-hand side. The removal of this term was already conjectured in [Reference Tao and Teräväinen25, Remark 5.4].

Proof. Without loss of generality, we may assume that $z \leq w$ . Expanding out the expression $\| \Lambda _{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w} - \Lambda _{\mathrm{Cram}\acute{\rm e}\mathrm{r}, z} \|_{U^{d+1}(I)}^{2^{d+1}}$ into an alternating sum of $2^{d+1}$ terms, it suffices to show that

for all choices of parameters $w_\epsilon \in \{w,z\}$ , where $\epsilon = (\epsilon _1,\ldots ,\epsilon _{d+1})$ and X is a quantity that is independent of the choice of parameters $w_\epsilon $ . Applying Proposition 4.3, the left-hand side is

$$ \begin{align*} \mathrm{vol}(\Omega) \prod_p \beta_p + O_d( N^{d+2} \exp(-c \operatorname{Log}^{4/5} N)),\end{align*} $$

where $\Omega $ is a certain explicit convex polytope of volume $\beta _\infty N^{d+2}$ for some constant $\beta _\infty $ depending only on d, and the local factors $\beta _p$ are defined by the formula

The local factors $\beta _p$ are independent of the $w_\epsilon $ if $p \leq w$ or $p> z$ . Thus, by equation (4.4), the product $\prod _p \beta _p$ can be written as $Y(1+O(1/z))$ for some Y that is independent of the $w_\epsilon $ parameters, and the claim follows.

4.2 Bounds on the Heath-Brown approximant

We now turn to the Heath-Brown approximants $\Lambda _{\operatorname {HB},Q}$ . The nice bounds in $\ell ^\infty $ or $\ell ^1$ one has in Lemma 4.1 or Corollary 4.4 are unfortunately not available for this approximant. However, we have reasonable control in other norms such as $\ell ^2$ , in large part due to a good Type I representation.

Lemma 4.6. (Moment bounds for Heath-Brown approximant)

For any $Q \geq 1$ , one has the Type I representation

(4.5)

for some weights $\unicode{x3bb} _{\ell }$ with

(4.6)

In particular, we have the pointwise bound

(4.7) $$ \begin{align} \Lambda_Q(n) \lesssim \tau(n,Q) \langle \operatorname{Log} Q \rangle, \end{align} $$

where $\tau (n,Q)$ is the truncated divisor function

Furthermore, we have the moment bounds

(4.8) $$ \begin{align} \mathbb{E}_{n \in [N]} |\Lambda_Q(n)|^k \lesssim_k \langle \operatorname{Log} Q\rangle^{2^k+k} \end{align} $$

for any positive integer k and $N \geq 1$ .

Proof. Applying the standard identity $c_q(n)=\sum _{\ell \mid (q,n)}\ell \mu (q/\ell )$ and then writing $q=\ell r$ , we have

$$ \begin{align*} \Lambda_Q(n) &= \sum_{q<Q} \frac{\mu(q)}{\varphi(q)}\sum_{\ell\mid (q,n)}\ell\mu(q/\ell) \\ &= \sum_{\substack{\ell\mid n\\ \ell<Q}} \frac{\mu(\ell) \ell }{\varphi(\ell)} \sum_{\substack{r < Q/\ell\\ (\ell,r)=1}} \frac{\mu^2(r)}{\varphi(r)}. \end{align*} $$

We then take

From Rankin’s trick and Mertens’s theorem, for any $1\leq d\leq Q$ , one has

$$ \begin{align*} \sum_{\substack{r \leq Q/\ell\\ (d,\ell)=1}} \frac{\mu^2(r)}{\varphi(r)} &\lesssim \sum_{\substack{r\geq 1\\ (\ell,r)=1}} \frac{\mu^2(r)}{\varphi(r) r^{1/\langle \operatorname{Log} Q\rangle}} \\ &\lesssim \prod_{\substack{p\\ p \nmid \ell}} \bigg(1+\frac{1}{(p-1) p^{1/\langle \operatorname{Log} Q\rangle}}\bigg) \\ &\lesssim \frac{\varphi(\ell)}{\ell} \prod_p \bigg(1 + \frac{1}{p^{1+1/\langle \operatorname{Log} Q\rangle}} + O\bigg(\frac{1}{p^2}\bigg)\bigg) \\ &\lesssim \frac{\varphi(\ell)}{\ell} \langle \operatorname{Log} Q\rangle, \end{align*} $$

where we used the Euler product formula and the standard bound $\zeta (\sigma )\sim {1}/{(\sigma -1)}$ for $\sigma>1$ to estimate the product over the primes. This gives equation (4.6). The bound in equation (4.7) then follows from the triangle inequality.

Now, we turn to equation (4.8). We may assume that $Q \geq 100$ , as the claim is trivial otherwise. We allow all implied constants to depend on k. In view of equation (4.7), it suffices to establish the bound

$$ \begin{align*} \sum_{n \in [N]} \tau(n,Q)^k \lesssim N \langle \operatorname{Log} Q\rangle^{2^k}.\end{align*} $$

We expand

$$ \begin{align*} \sum_{n \in [N]} \tau(n,Q)^k = \sum_{n \in [N]} \bigg(\sum_{\substack{\ell\mid n\\d<Q}}1\bigg)^k = \sum_{n \in [N]} \sum_{\ell_1,\ldots, \ell_k<Q} 1 = \sum_{\ell_1,\ldots, \ell_k<Q} \frac{N}{[\ell_1,\ldots, \ell_k]}, \end{align*} $$

where $[a_1,\ldots , a_k]$ is the least common multiple of $a_1,\ldots , a_k$ .

Now, we apply Rankin’s trick. For $\ell _i<Q$ , we have $\ell _i^{1/\langle \operatorname {Log} Q\rangle } = O(1)$ , and thus,

$$ \begin{align*} \mathbb{E}_{n \in [N]} \tau(n,Q)^k &\lesssim \sum_{\ell_1,\ldots, \ell_k} \frac{1}{\ell_1^{1/\log Q}\cdots \ell_k^{1/\langle \operatorname{Log} Q\rangle}[\ell_1,\ldots, \ell_k]}. \end{align*} $$

Factorizing into an Euler product, we conclude that

$$ \begin{align*} \mathbb{E}_{n \in [N]} \tau(n,Q)^k \lesssim \prod_{p}\bigg(1+\sum_{\substack{a_1,\ldots, a_k\in \{0,1\}\\ (a_1,\ldots,a_k)\neq \mathbf{0}}}\frac{1}{p^{1+(a_1+\cdots a_k)/\langle \operatorname{Log} Q\rangle}} +O\bigg(\frac{1}{p^2}\bigg)\bigg), \end{align*} $$

where . Hence, on taking logarithms, it will suffice to show that

$$ \begin{align*} \sum_p \sum_{\substack{a_1,\ldots,a_k\in \{0,1\}\\ (a_1,\ldots, a_k)\neq \mathbf{0}}}p^{-1- ({a_1+\cdots+ a_k})/{\langle \operatorname{Log} Q\rangle}} \leq 2^k \log\log Q + O(1). \end{align*} $$

From partial summation and the prime number theorem, we have

$$ \begin{align*} &\sum_{\substack{a_1,\ldots,a_k\in \{0,1\}\\ (a_1,\ldots, a_k)\neq \mathbf{0}}}\sum_{p\geq Q}p^{-1- ({a_1+\cdots+ a_k})/{\langle \log Q\rangle}}\\ &\quad\leq \sum_{\substack{a_1,\ldots,a_k\in \{0,1\}\\ (a_1,\ldots, a_k)\neq \mathbf{0}}} \int_{Q}^{\infty}\frac{t^{-1-({a_1+\cdots+a_k})/{\langle \operatorname{Log} Q\rangle}}}{\log t}\, dt+O(1)\\ &\quad\leq 2^k \cdot \int_{Q}^{\infty} t^{-{1}/{\langle \log Q\rangle}} \ \frac{dt}{t \log t} +O(1) \lesssim 2^k + O(1). \end{align*} $$

Moreover, we can use Mertens’s theorem to estimate

$$ \begin{align*} \sum_{\substack{a_1,\ldots,a_k\in \{0,1\}\\ (a_1,\ldots, a_k)\neq \mathbf{0}}}\sum_{p<Q}p^{-1- ({a_1+\cdots+ a_k})/{\langle \operatorname{Log} Q\rangle}} \leq 2^k\log \langle \operatorname{Log} Q \rangle +O(1). \end{align*} $$

Combining these bounds gives the result.

4.3 Comparing the Cramér and Heath-Brown approximants

We have a useful comparison theorem between the Cramér and Heath-Brown approximants.

Proposition 4.7. (Comparison between Cramér and Heath-Brown)

Let $N \geq 1$ and $1 \leq w, Q \leq \exp (\operatorname {Log}^{1/20} N)$ , and let $d \geq 1$ be an integer. Then,

$$ \begin{align*} \| \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w} - \Lambda_{\operatorname{HB},Q} \|_{u^{d+1}(I)} \lesssim_d w^{-c} + Q^{-c}\end{align*} $$

for any interval I of length N. As a consequence, from Lemma 4.5 and the triangle inequality, we also have

$$ \begin{align*} \| \Lambda_{\operatorname{HB},Q_1} - \Lambda_{\operatorname{HB},Q_2} \|_{u^{d+1}(I)} \lesssim_d Q_1^{-c} + Q_2^{-c}\end{align*} $$

whenever $1 \leq Q_1,Q_2 \leq \exp (\operatorname {Log}^{1/20} N)$ .

Proof. We allow all implied constants to depend on d. In view of Lemma 4.5 and the triangle inequality, it suffices to establish the bound

$$ \begin{align*} \| \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r},Q} - \Lambda_{\operatorname{HB},Q} \|_{u^{d+1}(I)} \lesssim Q^{-c}\end{align*} $$

for any interval I of length N, that is to say, it suffices to show that

$$ \begin{align*} |\mathbb{E}_{n \in I} (\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r},Q}(n) - \Lambda_{\operatorname{HB},Q}(n)) e(R(n)) | \lesssim Q^{-c}\end{align*} $$

for any polynomial $R(n) = \sum _{j=0}^d \alpha _j (n-n_I)^d$ of degree at most d with some real coefficients $\alpha _j$ , where $n_I$ denotes the midpoint of I. By subdividing I into smaller intervals and using the triangle inequality (adjusting the coefficients $\alpha _j$ as necessary), we may assume without loss of generality that

$$ \begin{align*} N \sim \exp(\operatorname{Log}^{20} Q).\end{align*} $$

We can then also assume that Q (and hence N) are large, as the claim is trivial otherwise. In particular, $\operatorname {Log} N = \operatorname {Log}^{O(1)} Q$ , which in practice will permit us to absorb all logarithmic factors of N in the analysis below.

Fix the polynomial R. We may of course assume without loss of generality that

$$ \begin{align*} |\mathbb{E}_{n \in I} (\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r},Q}(n) - \Lambda_{\operatorname{HB},Q}(n)) e(R(n)) |\geq Q^{-1}.\end{align*} $$

Applying Lemma 4.2 (with $w=Q$ and $y = \exp (\operatorname {Log}^{1/10} N)$ ) as well as Lemma 4.6, we thus have

for some weights $\unicode{x3bb} _{\ell }$ of size $O(\operatorname {Log}^{O(1)} N) = O(\operatorname {Log}^{O(1)} Q)$ . Applying [Reference Matomäki and Shao17, Proposition 2.1] (after shifting the summation variable by $n_I$ ), we conclude that the polynomial R is major arc in the sense that there exists an integer $1 \leq q \lesssim Q^{O(1)}$ such that

$$ \begin{align*} \| q \alpha_j \|_{\mathbb{R}/\mathbb{Z}} \lesssim Q^{O(1)} / N^j\end{align*} $$

for all $1 \leq j \leq d$ . We may assume that $q\geq Q$ by multiplying q by an integer of size Q if necessary. Thus, one can write $R(n) = R_0(n) + E(n)$ , where $R_0$ is a polynomial of degree at most d that is periodic with period q and the error E satisfies $\sup _{n\in I}|E(n+1)-E(n)|=O( Q^{O(1)}/N)$ .

Set

and thus, $Q \leq w \lesssim Q^{O(1)}$ . By Lemma 4.5 and the triangle inequality, it will suffice to show that

$$ \begin{align*} |\mathbb{E}_{n \in I} (\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w}(n) - \Lambda_{\operatorname{HB},Q}(n)) e(R(n)) | \lesssim Q^{-c}.\end{align*} $$

Breaking up I into intervals J of length $\sqrt {N}$ and using the slowly varying nature of $E(n)$ , it suffices to show that

$$ \begin{align*} |\mathbb{E}_{n \in J} (\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w}(n) - \Lambda_{\operatorname{HB},Q}(n)) e(R_0(n)) |\lesssim Q^{-c}\end{align*} $$

for any interval J of length $\sqrt {N}$ .

From Corollary 4.4 and the q-periodicity of $R_0$ , we have

$$ \begin{align*} \mathbb{E}_{n \in J} \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w}(n) e(R_0(n)) = \mathbb{E}_{n \in (\mathbb{Z}/q\mathbb{Z})^\times} e(R_0(n)) + O(Q^{-c})\end{align*} $$

(in fact, the error term is significantly better than this). Using the multiplicativity of the Ramanujan sums $c_q(\cdot )$ and the fact that , we have

We thus have

$$ \begin{align*} \mathbb{E}_{n \in J} \Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w}(n) e(R_0(n)) = \sum_{\ell\mid q} \frac{\mu(\ell)}{\varphi(\ell)} \mathbb{E}_{n \in [q]} e(R_0(n)) c_{\ell}(n) + O(Q^{-c}).\end{align*} $$

Note that for any natural numbers $\ell ,a,q$ with $\ell \nmid q$ , by the geometric sum formula, we have

$$ \begin{align*} \mathbb{E}_{n\in J}c_{\ell}(n)1_{n\equiv a\pmod q}=\sum_{r\in (\mathbb{Z}/\ell\mathbb{Z})^{\times}}\mathbb{E}_{n\in J}e\bigg(\frac{rn}{\ell}\bigg)1_{n\equiv a\pmod q}\ll \ell^2/\sqrt{N}. \end{align*} $$

Therefore, from equation (4.1) and the q-periodicity of $e(R_0(n))$ , we have

$$ \begin{align*} \mathbb{E}_{n \in J} \Lambda_{\operatorname{HB},Q}(n) e(R_0(n)) = \sum_{\substack{\ell\mid q\\ \ell < Q}} \frac{\mu(\ell)}{\varphi(\ell)} \mathbb{E}_{n \in [q]} e(R_0(n)) c_{\ell}(n) + O(Q^{-c})\end{align*} $$

(again, a better error term is available here). Thus, by the triangle inequality, it suffices to show that

$$ \begin{align*} \sum_{\substack{\ell\mid q\\ \ell\geq Q}} \frac{\mu^2(\ell)}{\varphi(\ell)} |\mathbb{E}_{n \in [q]} e(R_0(n)) c_{\ell}(n)| \lesssim Q^{-c}.\end{align*} $$

By the divisor bound, q has at most $Q^{o(1)}$ factors, so it will suffice to establish the bound

$$ \begin{align*} |\mathbb{E}_{n \in [q]} e(R_0(n)) c_{\ell}(n)| \lesssim \varphi(\ell) Q^{-c}\end{align*} $$

for each square-free $\ell \mid q$ with $\ell \geq Q$ . By the triangle inequality, it suffices to show that

$$ \begin{align*} \sum_{r \in (\mathbb{Z}/\ell\mathbb{Z})^\times} |\mathbb{E}_{n \in \mathbb{Z}/q\mathbb{Z}} e(R_0(n) - rn/\ell)| \lesssim \varphi(\ell) Q^{-c}.\end{align*} $$

However, from the Plancherel identity (or Bessel inequality) and the fact that $\ell \leq q$ , one has

$$ \begin{align*} \sum_{r \in (\mathbb{Z}/\ell\mathbb{Z})^\times} |\mathbb{E}_{n \in \mathbb{Z}/q\mathbb{Z}} e(R_0(n) - rn/\ell)|^2 \leq \frac{\ell}{q}\leq 1,\end{align*} $$

and the claim follows from Cauchy–Schwarz (noting from the hypothesis $\ell \geq Q$ that $\varphi (\ell ) \gtrsim Q^{1/2}$ , say, so that $\varphi (\ell )^{1/2} \lesssim \varphi (\ell ) Q^{-1/4}$ ).

6.1 Peluse’s inverse theorem for the primes

As is clear from the previous sections, Peluse’s inverse theorem [Reference Peluse23] was an important ingredient in the proof of the unweighted bilinear ergodic theorem in [Reference Krause, Mirek and Tao13]. In the course of proving Theorem 1.3, we essentially needed a version of this inverse theorem where one of the variables was weighted by the approximant $\Lambda _N$ ; see Proposition 5.3. It is natural to ask if one can also obtain a version of Peluse’s inverse theorem with the von Mangoldt weight $\Lambda $ . We record here how such a result quickly follows from the arguments used to prove Proposition 5.3.

Theorem 6.1. (Peluse’s inverse theorem with prime weight)

Let $k,d\in \mathbb {N}$ and $A>0$ . Let $N\geq 2$ , $(\log N)^{-A}\leq \delta \leq 1$ and $N_0\sim N^d$ . Let $P_1,\ldots , P_k$ be polynomials with integer coefficients of distinct degrees, with maximal degree d. Let $h,f_1,\ldots , f_k\colon \mathbb {Z}\to \mathbb {C}$ be functions bounded in modulus by $1$ and supported on $[-N_0,N_0]$ . Suppose that

(6.1) $$ \begin{align} \bigg|\!\sum_{x\in \mathbb{Z}}\mathbb{E}_{n\in [N]}\Lambda(n)h(x)f_1(x+P_1(n))\cdots f_k(x+P_k(n))\bigg| \geq \delta N^d. \end{align} $$

Then, either $N_0\lesssim _{P_1,\ldots , P_k} \delta ^{-O_d(1)}$ or there exists a positive integer $q\lesssim _{P_1,\ldots , P_k} \delta ^{-O_d(1)}$ and $\delta ^{O_d(1)}N\lesssim _{P_1,\ldots , P_k} N'\leq N$ such that

$$ \begin{align*} \frac{1}{N^d}\bigg|\!\sum_{x\in \mathbb{Z}}\mathbb{E}_{m\in [N']}f_1(x+qm)\bigg|\gtrsim_{A,P_1,\ldots, P_k}\delta^{O_d(1)}. \end{align*} $$

Proof. Fix $P_1,\ldots , P_k$ ; we allow all implied constants to depend on them. Define the polynomial averaging operator

Let $w_0=\delta ^{-C_d}$ for a large enough constant $C_d$ . We claim that

(6.2) $$ \begin{align} T_{N,\Lambda-\Lambda_N}(h,f_1,\ldots, f_k)\lesssim_A (\log N)^{-A}, \end{align} $$

and

(6.3) $$ \begin{align} T_{N,\Lambda_N-\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r},w_0}}(h,f_1,\ldots, f_k)\lesssim \delta^2 \end{align} $$

and

(6.4) $$ \begin{align} T_{N,\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r},w_0}-\Lambda_{\operatorname{HB},w_0}}(h,f_1,\ldots, f_k)\lesssim \delta^2. \end{align} $$

After we have these three estimates, we conclude from equation (6.1) and linearity that

$$ \begin{align*} |T_{N,\Lambda_{\operatorname{HB},w_0}}(h,f_1,\ldots, f_k)|\gtrsim \delta. \end{align*} $$

By equations (4.1) and (4.2), the function $\Lambda _{\operatorname {HB},w_0}$ is a linear combination, with $1$ -bounded coefficients, of $O(w_0^3)$ indicators of arithmetic progressions of common difference at most $w_0$ . Hence, crudely using the triangle inequality, we obtain

for some $1\leq a\leq q'\lesssim \delta ^{-O_d(1)}$ . However, now the claim of the theorem follows from [Reference Peluse23, Theorem 3.3] after making a change of variables.

We are left with showing equations (6.2), (6.3) and (6.4). The estimate in equation (6.2) follows immediately from [Reference Teräväinen26, Theorem 4.1] and equation (3.1). The estimate in equation (6.3) follows by using Lemmas 4.5, 4.1 and [Reference Teräväinen26, Theorem 4.1] to obtain

$$ \begin{align*} T_{N,\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w}-\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, z}}(h,f_1,\ldots, f_k)\lesssim w^{-c_d} \end{align*} $$

for some $c_d>0$ and any $z\in [w/2,w]$ , $1\leq w\leq \exp ((\log N)^{1/10})$ , and then summing this dyadically. For proving equation (6.4), note that from equation (5.14) and [Reference Teräväinen26, Theorem 4.1], we have for any $\kappa>0, \varepsilon >0$ , the bound

$$ \begin{align*} T_{N,\Lambda_{\mathrm{Cram}\acute{\rm e}\mathrm{r}, w_0}-\Lambda^{\prime}_{\operatorname{HB},w_0}}(h,f_1,\ldots, f_k) \lesssim_\varepsilon \langle \operatorname{Log} w_0\rangle^{O_{\varepsilon}(1)}(\kappa^{c_d^{\prime}} + \kappa^{-\varepsilon} w_0^{-c_d^{\prime}}) N^d, \end{align*} $$

with $\Lambda ^{\prime }_{\operatorname {HB},w_0}$ obeying equation (5.13). However, from equation (5.13) and the triangle inequality, we now obtain equation (6.4) by taking $\varepsilon>0$ small enough and $\kappa =w_0^{-c}$ for a small enough constant c (depending on d). This was enough to complete the proof.

6.2 Siegel zeroes

In this subsection, we mention an alternative approach to Theorem 1.3 based on working with Siegel zeroes. This approach is somewhat more complicated than that implemented above and we shall only sketch it very briefly, leaving the details to the interested reader.

The place in the proof of Theorem 1.3 where passing from the von Mangoldt function $\Lambda $ to the approximant $\Lambda _N$ avoided dealing with Siegel zeroes is Proposition 3.4, so we begin by sketching how a variant of Proposition 3.4 can be proven for the weight $\Lambda $ .

We say that a modulus $q\geq 2$ is exceptional if there exists a non-principal real Dirichlet character $\chi _q\pmod q$ such that $L(s,\chi _q)$ has a real zero $\beta _q>1-c_0/(\log q)$ , where $c_0$ is some small absolute constant. We call the corresponding character $\chi _q$ an exceptional character and we call $\beta _q$ a Siegel zero. For any given exceptional q, the character $\chi _q$ and Siegel zero $\beta _q$ are uniquely determined.

For exceptional characters $\chi _q$ , we define the arithmetic symbol

and the (weighted) continuous multiplier

where $\beta _q\in (0,1)$ is the Siegel zero. Then, if we replace in equation (3.18),

$$ \begin{align*} &\mathrm{B}^{l_1, l_2, m_{\hat {\mathbb{Z}}^\times}}_{(\eta_{\leq -\operatorname{Log} N+s} \otimes \eta_{\leq -d\operatorname{Log} N+ds})\tilde m_{N,\mathbb{R}} } \\ & \qquad \longrightarrow \mathrm{B}^{l_1, l_2, m_{\hat {\mathbb{Z}}^\times}}_{(\eta_{\leq -\operatorname{Log} N+s} \otimes \eta_{\leq -d\operatorname{Log} N+ds})\tilde m_{N,\mathbb{R}} } + \sum_{q \text{ exceptional}} \mathrm{B}^{l_1, l_2, m_{\hat {\mathbb{Z}}^\times,\chi_q}}_{(\eta_{\leq -\operatorname{Log} N+s} \otimes \eta_{\leq -d\operatorname{Log} N+ds})\tilde m_{N,\mathbb{R},\chi_q} }, \end{align*} $$

the conclusion of Proposition 3.4 holds with the von Mangoldt weight $\Lambda $ in place of $\Lambda _N$ . This follows from essentially the same proof as in §5, but using the Landau–Page theorem [Reference Montgomery and Vaughan20, Corollary 11.10] in place of Corollary 4.4.

In the large-scale regime, the error bounds arising from the Siegel–Walfisz theorem remove the need for the above approximation; in the small-scale regime,

$$ \begin{align*}\{ N \in \mathbb{D}\colon 2^{u^{O(1/(C_0 \rho))}} \leq N \leq 3^{C_0 \cdot 2^u} \},\end{align*} $$

further analysis is required to reduce matters to the two-parameter Rademacher–Menshov inequality.

The first observation is the classical fact that there is at most one exceptional character at each dyadic scale:

(6.5) $$ \begin{align} |\{q\in (2^j,2^{j+1}]\colon q \text{ exceptional} \}| \leq 1. \end{align} $$

We let $q_j$ denote the unique exceptional modulus in $(2^j,2^{j+1}]$ and abbreviate $\beta _j = \beta _{q_j}$ .

We then introduce a dyadic decomposition

$$ \begin{align*} \sum_{q \text{ exceptional}} \mathrm{B}^{l_1, l_2, m_{\hat {\mathbb{Z}}^\times,\chi_q}}_{(\eta_{\leq -\operatorname{Log} N+s} \otimes \eta_{\leq -d\operatorname{Log} N+ds})\tilde m_{N,\mathbb{R},\chi_q}} = \sum_{j \leq 2^{\rho l}} C_{N,j}(f,g),\end{align*} $$

where

$$ \begin{align*} &C_{N,j}(f,g)(x) \\ & = \int_{1/2}^1 \bigg( \int_{\mathbb{T}^2} \sum_{(a_1/q_j,a_2/q_j)\colon \mathrm{h}(a_i/q_j) = 2^{l_i}} m_{\hat{\mathbb{Z}}^{\times},\chi_{q_j}}(a_1/q_j,a_2/q_j) e(a_1 x/q_j + a_2 x/q_j) \\ & \quad\times ( \hat{f}(\xi_1 + a_1/q_j) \cdot \varphi(2^u \xi_1) \cdot e(\xi_1 N t) ) \\ & \quad\times ( \hat{g}(\xi_2 + a_2/q_j) \cdot \varphi(2^{du} \xi_2) \cdot e( \xi_2 P(Nt)) ) e(\xi_1 x + \xi_2 x) \cdot N^{\beta_{j} - 1} t^{\beta_{j} - 1} \ d\xi_1 d\xi_2 \bigg) \, dt. \end{align*} $$

The key novelty then derives from proving the following modified Rademacher– Menshov-type inequality, similar to [Reference Krause, Mirek and Tao13, Lemma 8.2].

Lemma 6.2. Let $V,W$ be normed vector spaces, $K,J$ be two positive integers and let $0<q<\infty $ . Let $B_j\colon V\times W\rightarrow L^q(X)$ be a family of bilinear operators for $j\in [J]$ . Let $\{f_{k}^j\}, \{g_k^j\}$ be sets of functions with $f_k^j\in V$ and $g_k^j\in W$ for $k\in [K]$ and $j\in [J]$ . Then,

$$ \begin{align*}&\bigg\|V^2\bigg(\sum_{j\in [J]}B_j(f_k^j,g_k^j)\colon k \in [K]\bigg)\bigg\|_{L^q(X)} \\ &\ \lesssim_{q} \langle \operatorname{Log} K\rangle^{O_q(1)} \kern-3pt\sup_{\epsilon_{k}^{j}, {\varepsilon}_k^j \in \{ \pm 1 \}}\kern-1pt \bigg\|\kern-1pt \sum_{j\in [J]}\kern-2pt B_j\bigg(\kern-1pt\sum_{k\in [K]} \epsilon_k^j (f_k^j- f_{k-1}^j), \kern-2pt\sum_{k\in [K]}\kern-1.3pt {\varepsilon}_k^j (g_k^j- g_{k-1}^j)\kern-1.2pt\bigg)\bigg\|_{L^q(X)}\!. \end{align*} $$

This result may be of independent interest, so we provide a brief proof.

Proof. Set $a_{k_1,k_2}= \sum _{j \in [J]} B_j(f_{k_1}^j,g_{k_2}^j)$ . By [Reference Krause, Mirek and Tao13, Lemma 8.1], we have

$$ \begin{align*} V^2\bigg(\sum_{j \in [J]} B_j(f_k^j,g_k^j)\colon k\in [K]\bigg) \lesssim \sum_{\substack{M_1,M_2<K\\ \mathcal{M}_1,M_2\colon \text{dyadic}}} \bigg\| \Delta \sum_{j \leq J} B_j(f_{M_1n_1}^j,g_{M_2n_2}^j)\bigg\|_{\ell^2(n_1,n_2)}, \end{align*} $$

where

$$ \begin{align*} \Delta \sum_{j \in [J]} B_j(f_{M_1n_1}^j,g_{M_2n_2}^j) =& \sum_{ j \in [J]} B_j(f_{M_1n_1}^j,g_{M_2n_2}^j) - \sum_{j\in [J]} B_j(f_{(n_1-1)M_1}^j,g_{M_2n_2}^j)\\ &- \sum_{j \in [J]} B_j(f_{M_1n_1}^j,g_{(n_2-1)M_2}^j) + \kern-2pt\sum_{j \in [J]}\kern-2pt B_j(f_{(n_1-1)M_1}^j,g_{(n_2-1)M_2}^j). \end{align*} $$

Taking

$$ \begin{align*} \tilde f_{M_1n_1} = f_{M_1n_1} - f_{(n_1-1)M_1},\quad \tilde g_{M_2n_2} = g_{M_2n_2} - g_{(n_2-1)M_2},\end{align*} $$

we need to bound

(6.6) $$ \begin{align} \langle \operatorname{Log} K\rangle^{O_q(1)} \sup_{M_1,M_2 < K \text{ dyadic}} \bigg\| \bigg(\sum_{\substack{n_1<k/M_1\\ \mathrm{n}_2<k/M_2}}\bigg| \sum_{j\in [J]} B_j(\tilde f_{M_1n_1}^j,\tilde g_{M_2n_2}^j)\bigg|^2\bigg)^{1/2}\bigg\|_{L^q(X)}. \end{align} $$

Applying Khintchine’s inequality

$$ \begin{align*} \bigg( \sum_{n} |a_n|^2 \bigg)^{1/2} = \bigg( \mathbb{E}_{\epsilon_n \in \pm 1}\bigg| \sum_n \epsilon_n a_n\bigg|^2 \bigg)^{1/2} \sim_q \bigg( \mathbb{E}_{\epsilon_n \in \pm 1}\bigg| \sum_n \epsilon_n a_n\bigg|^q \bigg)^{1/q}, \end{align*} $$

we arrive at the following chain of inequalities:

$$ \begin{align*} &\bigg\| V^2\bigg(\sum_{j\in [J]}B_j(f_k^j,g_k^j)\colon k \in [K]\bigg)\bigg\|_{L^q(X)} \\ &\ \ \lesssim \langle \operatorname{Log} K\rangle^{O_q(1)} \sup_{M_1,M_2} \bigg\| \bigg(\mathbb{E}_{\varepsilon_{n_2} \in \pm 1} \sum_{\substack{n_1}}\bigg| \sum_{n_2}\sum_{j\in [J]} \varepsilon_{n_2} B_s(\tilde f_{M_1n_1}^j,\tilde g_{M_2n_2}^j)\bigg|^2\bigg)^{1/2}\bigg\|_{L^q(X)} \\ &\ \ \lesssim \langle \operatorname{Log} K\rangle^{O_q(1)} \sup_{M_1,M_2} \bigg\| \bigg(\mathbb{E}_{\epsilon_{n_1}, \varepsilon_{n_2} \in \pm 1}\bigg| \sum_{\substack{n_1}} \sum_{n_2}\sum_{j\in [J]} \epsilon_{n_1} \varepsilon_{n_2} B_j(\tilde f_{M_1n_1}^j,\tilde g_{M_2n_2}^j)\bigg|^2\bigg)^{1/2}\bigg\|_{L^q(X)} \\ &\ \ \lesssim_q \langle \operatorname{Log} K\rangle^{O_q(1)} \kern-2pt\sup_{M_1,M_2}\kern-1pt \bigg\| \bigg(\mathbb{E}_{\epsilon_{n_1}, \varepsilon_{n_2} \in \pm 1} \bigg|\sum_{\substack{n_1}} \sum_{n_2}\kern-2pt\sum_{j\in [J]} \epsilon_{n_1} \varepsilon_{n_2} B_j(\tilde f_{M_1n_1}^j,\tilde g_{M_2n_2}^j)\bigg|^q\bigg)^{1/q}\bigg\|_{L^q(X)} \\ &\ \ \lesssim_q \langle \operatorname{Log} K\rangle^{O_q(1)} \sup_{M_1,M_2, \epsilon_{n_1},\epsilon_{n_2}} \bigg\| \sum_{\substack{n_1}} \sum_{n_2}\sum_{j\in [J]} \epsilon_{n_1} \varepsilon_{n_2} B_j(\tilde f_{M_1n_1}^j,\tilde g_{M_2n_2}^j)\bigg\|_{L^q(X)}. \end{align*} $$

By bilinearity, we may consolidate

$$ \begin{align*} \sum_{\substack{n_1}} \sum_{n_2}\sum_{j\in [J]} \epsilon_{n_1} \varepsilon_{n_2} B_j(\tilde f_{M_1n_1}^j,\tilde g_{M_2n_2}^j) = \sum_{j\in [J]} B_j\bigg(\sum_{n_1} \epsilon_{n_1} \tilde f_{M_1 n_1}^j, \sum_{n_2} \varepsilon_{n_2}\tilde g_{M_2n_2}^j\bigg);\end{align*} $$

putting everything together,

$$ \begin{align*} &\bigg\| {V}^2\bigg(\sum_{j\in [J] }B_j(f_k^j,g_k^j)\colon k \in [K]\bigg)\bigg\|_{L^q(X)} \\ &\quad\lesssim \langle \operatorname{Log} K\rangle^{O_q(1)} \sup_{\substack{M_1,M_2\\ \epsilon_{n_1}, \varepsilon_{n_2}}} \bigg\| \sum_{j \in [J]} B_j\bigg( \sum_{n_1} \epsilon_{n_1} ( f_{M_1n_1}^j- f_{(n_1-1)M_1}^j),\\ &\quad\qquad\sum_{n_2}\varepsilon_{n_2} (g_{M_2n_2}^j-g_{(n_2-1)M_2}^j)\bigg)\bigg\|_{L^q(X)}, \end{align*} $$

and so we get the result upon telescoping e.g.

6.3 Breaking duality

We briefly remark that one may establish Theorem 1.3 with r-variation restricted to the range $r> 2 + \epsilon $ for exponents $p_1,p_2>1$ that satisfy

where $\epsilon '> 0$ is sufficiently small in terms of $\epsilon $ ; hence, going beyond the duality range.

The single-scale estimate

(6.7) $$ \begin{align} \| A_{N;\Lambda;X}(f,g) \|_{L^p(X)} \lesssim \| f \|_{L^{p_1}(X)} \| g \|_{L^{p_2}(X)} \end{align} $$

anchors the argument; equation (6.7) follows from Hölder’s inequality and the improving estimate Lemma 5.1, as per [Reference Krause, Mirek and Tao13, Lemma 11.1]. With equation (6.7) in hand, the proof of [Reference Krause, Mirek and Tao13, Proposition 11.4] can be formally reproduced, with only notational changes arising. We leave the details to the interested reader.

6.4 Sharpness of the variational result

The unboundedness of the quadratic variation along polynomial orbits, namely [Reference Krause, Mirek and Tao13, Proposition 12.1], extends to our context.

Proposition 6.3. Let $P \in \mathbb {Z}[\mathrm {n}]$ be a non-constant polynomial and let $0 < p \leq \infty $ . Let $I \subset \mathbb {N}$ be an infinite set. Then, for every $C> 0$ , there exists a measure-preserving system $(X,\mu ,T)$ of total measure 1 and a $1$ -bounded $f \in L^{\infty }(X)$ so that

$$ \begin{align*} \| ( \mathbb{E}_{p \in [N]} T^{P(p)} f )_{N \in I} \|_{L^p(X;V^2)} \geq C. \end{align*} $$

We shall leave the details of the proof of this proposition to the interested reader as it is similar to the proof of [Reference Krause, Mirek and Tao13, Proposition 12.1]. The key additional observation is the equidistribution of

$$ \begin{align*} p \mapsto ( \alpha_1 \cdot P(p),\ldots,\alpha_K \cdot P(p) ) \subset \mathbb{T}^K \end{align*} $$

over the primes whenever $\alpha _1,\ldots ,\alpha _K$ are $\mathbb {Q}$ -linearly independent and $P \in \mathbb {Z}[\mathrm {n}]$ is a non-constant polynomial (which follows from Weyl’s criterion and a standard exponential sum estimate for polynomials of primes; see e.g. [Reference Matomäki and Shao17, Theorem 1.3]).

To see why this implies the sharpness of the range of the variational estimate in Theorem 1.3, one may employ the convexity arguments of [Reference Mirek, Trojan and Zorin-Kranich19, §5], taking into account [Reference Mirek, Trojan and Zorin-Kranich19, Proposition 4.1], to obtain the lower bound

$$ \begin{align*} \| ( \mathbb{E}_{p \in [N]} T^{P(p)} f )_{N \in I} \|_{L^p(X;V^2)} \leq \| ( \mathbb{E}_{n \in [N]} \Lambda(n) \cdot T^{P(n)} f )_{N \in I} \|_{L^p(X;V^2)} + O(1). \end{align*} $$

6.5 Continuous extensions

From the perspective of density, the primes are ‘full/dimensional’, with a very ‘Fourier-uniform’ measure, $\Lambda $ . A natural question concerns establishing a continuous analogue of Theorem 1.3, namely the existence of a measure $\nu $ supported on $[0,1]$ , with (say) full Fourier dimension,

$$ \begin{align*} |\hat{\nu}(\xi)| \lesssim (1 + |\xi|)^{o(1)-1/2}\end{align*} $$

so that

$$ \begin{align*} \lim_{N \to \infty} \frac{1}{N} \int_0^N f(x-t) g(x-P(t)) \, d\nu(t), \quad d = \text{deg}(P) \geq 2 \end{align*} $$

exists almost everywhere whenever $f \in L^{p_1}(\mathbb {R})$ and $g \in L^{p_2}(\mathbb {R})$ with $p_1,p_2> 1$ and ${1}/{p_1} + {1}/{p_2} \leq 1$ . The key point is establishing a suitable Sobolev inequality, namely

$$ \begin{align*} \bigg\| \frac{1}{N} \int_0^N f(x-t) g(x-P(t)) \ d\nu(t) \bigg\|_{L^1([0,CN^d])} \lesssim (2^{-cl} + O_A(\langle \log N \rangle^{-A} )) N^d\end{align*} $$

for some $c> 0$ , whenever $|f|, |g| \leq 1$ , and $\hat {f}$ vanishes on $\{ |\xi | \lesssim 2^{l}/N \}$ and/or $\hat {g}$ vanishes on $\{ |\xi | \lesssim 2^l/N^d\}$ .

Estimates of this form in the unweighted setting go back to [Reference Bourgain1], with the strongest estimates recently established by one of us as part of a much more general phenomenon, see [Reference Krause, Mirek, Peluse and Wright12]. This approach relies on PET induction, which suggests that certain Gowers-uniformity conditions might need to be imposed on $\nu $ ; it is unclear how this might interact with dimension, so we leave the problem to the interested reader.