¹

1 Introduction↩︎

1.1 A brief history↩︎

A common phenomenon in the field of discrete analogs in harmonic analysis is that, for many important convolution operators with arithmetic features, their multipliers \(\mathfrak m(\xi)\) can be efficiently approximated by certain multifrequency expressions, which locally take the following form \[\begin{align} \label{eq:38} \mathfrak m_{\rm dis}(a/q) \cdot \mathfrak m_{\rm cont}(\xi - a/q) \end{align}\tag{1}\] if \(\xi\in\mathbb{T}\) lies close to a rational point \(a/q\) with a small denominator, and by \(0\) otherwise. Here \(\mathfrak m_{\rm dis}\) denotes the arithmetic component, typically evaluated at \(a/q\) as above, while \(\mathfrak m_{\rm cont}\) is the continuous component that reflects the behavior of the continuous prototype of the operator. One of the best-known examples is a discrete averaging Radon operator given by \[\begin{align} \label{eq:36} A_{N;\mathbb{Z}}^{P}f(x)\mathrel{\vcenter{:}}= \frac{1}{N}\sum_{n\in [N]}f(x-P(n)), \qquad x \in \mathbb{Z}, \end{align}\tag{2}\] for \(N\in\mathbb{Z}_+\) and a polynomial \(P\in\mathbb{Z}[{\rm n}]\) with integer coefficients, see Section 2 for notation.

Averages 2 were extensively studied by Bourgain [1]–[3] in the context of his celebrated pointwise ergodic theorem with polynomial iterates, see Theorem 6 in Section 6. In [1]–[3], Bourgain clearly emphasized that pointwise convergence problems in ergodic theory live in a very close symbiosis with quantitative estimates in the integer shift system, such as maximal or variational inequalities among others, see Sections 5 and 6. The latter inequalities ensure pointwise convergence in any measure-preserving system via the Calderón transference principle [4], which nowadays is the key tool linking the world of quantitative estimates in ergodic theory (see Theorem 7) with the world of discrete analogs in harmonic analysis, see the discussion in Section 6 below formula 56 .

An important fact is that the operator \(A_{N;\mathbb{Z}}^{P}\) is given by convolution with the kernel \[\begin{align} K_N^P\mathrel{\vcenter{:}}= \frac{1}{N}\sum_{n\in[N]}\mathbb{1}_{{\{P(n)\}}}. \end{align}\] Using the Fourier transform, see Section 2 for notation, we have \(\mathcal{F}_{\mathbb{Z}}A_{N; \mathbb{Z}}^{P} f(\xi)=m_{N}(\xi)\mathcal{F}_{\mathbb{Z}}f(\xi)\) for \(\xi\in\mathbb{T}\), where the multiplier \(m_N\) is a normalized exponential sum of the form \[\begin{align} \label{eq:37} m_{N}(\xi) \mathrel{\vcenter{:}}= \mathcal{F}_{\mathbb{Z}}K_N^P(\xi)= \frac{1}{N}\sum_{n\in[N]}e(-\xi\cdot P(n)). \end{align}\tag{3}\]

Bourgain [1]–[3] had an ingenious insight to use the circle method of Hardy–Littlewood–Ramanujan to analyze the multipliers from 3 and to make the approximation 1 rigorous. Namely, for \(N\ge 1\), we define a continuous counterpart of \(m_N\) by \[\begin{align} \mathfrak{m}_N(\xi) \mathrel{\vcenter{:}}= \int_{0}^1e(-\xi\cdot P(Nt)) \,{\rm d}t, \qquad \xi\in\mathbb{R}. \end{align}\] Also, for every \(a/q\in\mathbb{Q}\), we define the complete exponential sum by \(G(a/q) \mathrel{\vcenter{:}}= m_q(a/q)\). As a consequence of the mean-value theorem, for every \(\xi\in\mathbb{T}\) and \((a,q) \in\mathbb{Z} \times \mathbb{Z}_+\) satisfying \({\rm gcd}(a, q)=1\), if \(|\xi-a/q|\le N^{-\deg P+\delta}\) and \(q\le N^\delta\), then one has \[\begin{align} \label{eq:40} m_N(\xi)=G(a/q)\mathfrak m_N(\xi-a/q)+E_N(\xi), \end{align}\tag{4}\] where \(\|E_N\|_{L^\infty(\mathbb{T})}\le N^{-1/2}\) provided that \(\delta\in(0, 1)\) is sufficiently small.

Identity 4 is the key mechanism for constructing approximations on the major arcs in the classical circle method, thereby justifying 1 whenever \(\xi\in\mathbb{T}\) is in a small vicinity of some \(a/q\in\mathbb{Q}\) such that \(q \in \mathbb{Z}_+\) is small, i.e. \(|\xi-a/q|\le N^{-\deg P+\delta}\) and \(q\le N^\delta\). This was first observed by Bourgain in the context of discrete operators of Radon type 2 and their connections to pointwise ergodic theorems.

In order to gain a better perspective on the identity 4 and its role in the circle method, we refer to a vast literature in analytic number theory on the classical circle method; the books by Iwaniec and Kowalski [5], Nathanson [6], Vaughan [7], Vinogradov [8], and the notes of Wooley [9] are excellent references on this exciting subject.

Identity 4 , together with the classical Weyl inequality [10] for the exponential sums from 3 , led Bourgain [1]–[3] to construct an approximation for 3 , namely \[m_N(\xi)=\mathfrak a_N(\xi)+ E_N^*(\xi),\] where the error term \(E_N^*\) satisfies \(\|E_N^*\|_{L^\infty(\mathbb{T})}\le N^{-\gamma}\) for some small \(\gamma\in(0, 1)\) provided that \(\delta\in(0, 1)\) is sufficiently small, and \(\mathfrak a_N\) is a multifrequency multiplier given by the formula \[\begin{align} \label{eq:41} \mathfrak a_N(\xi)\mathrel{\vcenter{:}}=\sum_{a/q\in \mathcal{R}^1_{\le N^\delta}}G(a/q)\mathfrak m_N(\xi-a/q)\eta\big(N^{\deg P-\delta}(\xi-a/q)\big), \end{align}\tag{5}\] where \(\eta \colon \mathbb{R}\to[0, 1]\) is a smooth bump function so that \(\mathbb{1}_{{[-1/4, 1/4]}}\le \eta\le \mathbb{1}_{{(-1/2, 1/2)}}\), and \[\begin{align} \mathcal{R}_{\le N}^1 \mathrel{\vcenter{:}}= \big\{ a/q \in {\mathbb{Q}}/ \mathbb{Z}: q \in \{1,\dots,N\} \text{ and } {\rm gcd}(a, q)=1 \big \} \end{align}\] is the set of canonical fractions, where \(a/q\) is identified with its equivalence class \(a/q + \mathbb{Z}\).

The key outcome of this approximation is that the multiplier \(\mathfrak a_N\) is supported on the union of small disjoint neighborhoods of canonical fractions \(a/q \in \mathcal{R}_{\le N^\delta}^1\) provided that \(\delta\in(0, 1)\) is sufficiently small. Thus, it has precisely the form of the expression in 1 . By the disjointness of the summands in 5 and Plancherel’s theorem, the operator \(f \mapsto \mathcal{F}_{\mathbb{Z}}^{-1} (\mathfrak a_N \mathcal{F}_{\mathbb{Z}} f)\) is a contraction on \(\ell^2(\mathbb{Z})\) even though the cardinality of \(\mathcal{R}_{\le N^\delta}^1\) grows to infinity with \(N\). In fact, the latter is the main obstruction to extending the theory beyond \(\ell^2(\mathbb{Z})\).

After that reduction, the \(\ell^p(\mathbb{Z})\) bounds of maximal functions associated with 2 , or more generally other seminorms arising in the study of pointwise convergence problems, have been reduced to understanding the corresponding objects involving the operators \(f\mapsto \mathcal{F}_{\mathbb{Z}}^{-1}(\mathfrak a_N \mathcal{F}_{\mathbb{Z}} f)\) instead of \(A_{N; \mathbb{Z}}^{P} f\). However, the multifrequency nature of the multiplier \(\mathfrak a_N\) (i.e. \(\mathfrak a_N\) is a sum of pieces with disjoint supports centered at frequencies \(a/q\in\mathcal{R}_{\le N^\delta}^1\)) significantly complicates matters, even though each object, such as \[G(a/q)\quad \text{ or } \quad \mathfrak m_N(\xi-a/q)\eta\big(N^{\deg P -\delta}(\xi-a/q)\big),\] is well understood from the perspective of number theory and harmonic analysis for each individual \(a/q\in\mathcal{R}_{\le N^\delta}^1\). These complications are immediately apparent at the level of the \(\ell^p(\mathbb{Z})\) bounds for the operator \(f\mapsto \mathcal{F}_{\mathbb{Z}}^{-1}(\mathfrak a_N \mathcal{F}_{\mathbb{Z}} f)\) when \(p\neq 2\), even for each \(N\in\mathbb{Z}_+\) separately, since Plancherel’s theorem is not available on \(\ell^p(\mathbb{Z})\) spaces with \(p \neq 2\).

Bourgain instead developed a deep machinery in [1]–[3] to study discrete operators with multifrequency multipliers 5 . However, this approach required separating the arithmetic components \(G(a/q)\) from the continuous part \(\mathfrak m_N(\xi-a/q)\eta\big(N^{\deg P-\delta}(\xi-a/q)\big)\) by employing intricate methods from harmonic analysis and number theory, among others.

As far as we know, all available methods in the field of discrete analogs in harmonic analysis necessitate separate analysis of the arithmetic part and the continuous part. Our aim in this paper is to demonstrate that the discrete operators corresponding to multipliers like in 5 can be handled in a fairly unified way, without parceling out the arithmetic components from the continuous part, which will significantly simplify proof strategies.

1.2 Statement of the main results↩︎

More specifically, our aim is to prove a variant of the Ionescu–Wainger multiplier theorem [11] that incorporates arithmetic weights. Loosely speaking, this variant will allow us to work with discrete operators corresponding to multifrequency multipliers as in 5 and to analyze the arithmetic and continuous parts simultaneously. For this purpose, we introduce the new concept of Ionescu–Wainger weights. These weights will have certain algebraic properties, which ensure that the Ionescu–Wainger theory remains adaptable in the corresponding weighted setting. Here and throughout, we will use the basic notation introduced in Section 2.

For every Ionescu–Wainger weight \(G\), since \(G=\mathcal{F}_{(\mathbb{Z}/ q \mathbb{Z})^d}\mathcal{F}^{-1}_{(\mathbb{Z}/ q \mathbb{Z})^d} (G)\), we have \[\| G \|_{L^\infty(q^{-1}\mathbb{Z}^d)} \leq \| \mathcal{F}^{-1}_{(\mathbb{Z}/ q \mathbb{Z})^d} (G) \|_{L^1({(\mathbb{Z}/ q \mathbb{Z})^d})} \leq U_G.\] In particular, \(1 = G(0) \leq \sup_{\beta \in {\mathbb{Q}}^d} |G(\beta)| \leq U_G\) so that \({\boldsymbol{C}}_{G}(\Sigma)\le U_G^2\) by 6 .

Using the notation from 12 and 13 , we are now ready to state the first main result of this article, which concerns a multiplier as in 1 with an arithmetic component \(G\) and a continuous component \(\mathfrak m\), localized to the set of canonical fractions \(\mathcal{R}^d_{\le N}\).

Theorem 1 originates in the groundbreaking paper of Ionescu and Wainger [11]. The authors in [11] developed a deep multifrequency multiplier theorem in the spirit of Theorem 1 that allowed them (which was out of reach prior to [11]) to establish \(\ell^p(\mathbb{Z}^d)\) boundedness for all \(p\in(1, \infty)\) of the discrete singular Radon transforms \[\begin{align} \label{eq:45} R_N^Pf(x) \mathrel{\vcenter{:}}= \sum_{m\in \{-N, \dots, N\}^k\setminus\{0\}^k }f(x-P(m))K(m), \qquad x\in\mathbb{Z}^d, \end{align}\tag{9}\] where \(N\in\mathbb{Z}_+\) and \(P \colon \mathbb{Z}^k\to\mathbb{Z}^d\) is a polynomial mapping with integer coefficients, see item [itemc] in Example 1, and \(K\) is a Calderón–Zygmund kernel, see [11].

Ionescu and Wainger [11], assuming that for any arbitrary \(\rho\in(0, 1)\) the condition \[\begin{align} \label{eq:44} 0<\varepsilon < e^{-N^{\rho}} \end{align}\tag{10}\] is satisfied in place of condition 7 , originally proved Theorem 1 in the scalar setting, i.e. for \(H_1=H_2={\mathbb{C}}\), with \(G\equiv1\) and a bound \((\log N)^{2/\rho+1}\) in place of \(N^{{\boldsymbol{C}}_{\rm IW}(r) / \log \log N}\) in 8 , and for the so-called Ionescu–Wainger fractions in place of the canonical fractions \(\mathcal{R}^d_{\leq N}\) defined in 12 . The Ionescu–Wainger fractions are defined as follows \[\begin{align} \label{eq:43} \widetilde{\mathcal{R}}^d_{\leq N}\mathrel{\vcenter{:}}=\big\{ a/q \in({\mathbb{Q}}/ \mathbb{Z})^d: q \in P_N \text{ and } {\rm gcd} (a, q)=1\big\}, \end{align}\tag{11}\] where \(P_N\) is a certain large family of positive integers having delicate factorization properties that dictated condition 10 , which is much stronger than condition 7 .

Bourgain’s results [1]–[3] and the Ionescu–Wainger theorem [11] share the same starting point — the circle method — but otherwise have completely different proofs. Although [11] was a huge breakthrough, the full strength of the multifrequency multiplier theorem was not yet fully understood. The situation changed when the second author [12] highlighted that the Ionescu–Wainger multifrequency multiplier theorem can be interpreted as a multifrequency Littlewood–Paley theory that captures all the Diophantine features arising from the Hardy–Littlewood circle method. This gave a positive answer to Stein’s question about the existence of a discrete Littlewood–Paley theory and was used to obtain a new proof of the \(\ell^p(\mathbb{Z}^d)\) boundedness for all \(p\in(1, \infty)\) of the discrete singular Radon transforms 9 from [11]. The ideas from [12] led to many papers on the \(r\)-variational [13], [14], jump [15], or uniform oscillation [16] inequalities for discrete operators of Radon type.

Currently, the Ionescu–Wainger multifrequency multiplier theorem [11] is a key tool in the field of discrete analogs in harmonic analysis, and it has undergone a period of considerable changes and developments. As mentioned above, it was originally proved for the so-called Ionescu–Wainger fractions 11 . In [12], the second author, in addition to observing that the Ionescu–Wainger multiplier theorem can be interpreted as a multifrequency Littlewood–Paley theory, also proved that the norm is at most of order \(\log N\). In [15], the authors extended the Ionescu–Wainger multiplier theorem to the setting of arbitrary separable Hilbert spaces \(H_1\) and \(H_2\). This was valuable for applications to various square functions. In an important paper [17], Pierce undertook a successful effort to abstract the key orthogonality mechanisms behind the proof of the Ionescu–Wainger multiplier theorem in terms of the superorthogonality phenomena in harmonic analysis. This resulted in a very clean proof of the Ionescu–Wainger multiplier theorem. Essentially at the same time, Tao [18] provided an alternative proof of the Ionescu–Wainger multiplier theorem with bounds independent of \(N\). We emphasize that all these improvements to the Ionescu–Wainger multiplier theorem [12], [15], [17], [18] were achieved under the condition 10 for the set of Ionescu–Wainger fractions \(\widetilde{\mathcal{R}}^d_{\leq N}\) from 11 , as in the original Ionescu–Wainger approach [11].

Recently, the first two authors in collaboration with Peluse, Wan, and Wright [19], answered the question posed by Ionescu and Wainger in [11] in the affirmative, proving a variant of the Ionescu–Wainger multiplier theorem, as stated in Theorem 1 with \(G\equiv 1\) and the norm \(N^{{\boldsymbol{C}}_{\rm IW}(r) \log \log \log N / \log \log N}\) in place of \(N^{{\boldsymbol{C}}_{\rm IW}(r) / \log \log N}\) in 8 . This new Ionescu–Wainger multiplier theorem from [19] was essential in establishing a multilinear variant of Weyl’s inequality, which was key to Bergelson’s conjecture [19] for multilinear ergodic averages with commuting transformations for iterates along polynomials with distinct degrees. Theorem 1 can also be used to interpret the classical Weyl inequality for exponential sums [10] in \(\ell^p(\mathbb{Z})\) spaces for \(p \in (1, \infty)\). The details can be found in [20].

Now, having this brief outline of recent advances in the Ionescu–Wainger multiplier theorem, we can highlight the novelties in our results.

1. As far as we know, this is the first instance in the literature where the formulation of the Ionescu–Wainger multiplier theorem incorporates the arithmetic weights \(G\) satisfying conditions [cond:A] and [cond:B]. Surprisingly, the proof of the Ionescu–Wainger multiplier theorem with \(G \equiv 1\) from [19] is easily adaptable to our weighted setting without many differences. In Section 3, for the convenience of the reader, we present all the details, even those that were omitted in [19]. Thus, the price we pay for having a weighted version of the Ionescu–Wainger multiplier theorem, see Theorem 1 above, is not high, but the benefits from having such a result are substantial. Now the multipliers \(\llbracket G \mid \mathfrak m\rrbracket_{\mathcal{R}^d_{\le N}}\) defined in 13 can be handled in a fairly unified way without parceling out the arithmetic weights \(G\) from the continuous part \(\mathfrak m\).

2. In Section 5, we prove Theorem 5, which is a seminorm variant of Theorem 1, and will handle all seminorms arising in [13]–[16], see Definition 8. In Section 6, we will show how our weighted variants of the Ionescu–Wainger theory lead to unified and short proofs of all results from [13]–[16], see Theorem 7.

3. These weighted formulations of Theorem 1 and Theorem 5, together with the fact that they work with the set of canonical fractions \(\mathcal{R}^d_{\le N}\) from 12 , are among the most important outcomes of this paper that significantly simplify proof strategies. The first outcome avoids using complicated approximations handling the multipliers involving arithmetic components. The second avoids misalignment arising from working with the Ionescu–Wainger fractions \(\widetilde{\mathcal{R}}^d_{\le N}\) from 11 .

4. Here we also note that the norm in 8 is at most of order \(N^{\mathbf{C}_{\rm IW}(r)/\log\log N}\), which improves the upper bound \(N^{\mathbf{C}_{\rm IW}(r)\log\log\log N / \log\log N}\) from [19]. This is achieved by a slightly more delicate treatment of the divisor functions that arise in our argument. In fact, our bound \(N^{\mathbf{C}_{\rm IW}(r)/\log\log N}\) coincides with the upper bound for the divisor function. In Theorem 1, it is important that the norm in 8 is controlled by \(N^{\gamma}\) for any fixed \(\gamma\in(0,1)\), which covers all practical applications of inequality 8 that we are aware of. Of course the latter bound is implied by our upper bound.

5. A weighted analog of Theorem 1 for the Ionescu–Wainger fractions \(\widetilde{\mathcal{R}}^d_{\le N}\) in place of canonical fractions \(\mathcal{R}^d_{\le N}\) can be obtained by adapting the arguments from Tao’s paper [18]. As in [18], the implied bound in 8 is independent of \(N\).

6. Surprisingly, the dependence of the norm in inequality 8 for \(\mathcal{R}^d_{\le N}\) on the parameter \(N\) cannot be removed, which stands in sharp contrast to Tao’s result [18]. This provides a negative answer to [19], which was rather unexpected.

7. Although the bounds in 8 for \(\widetilde{\mathcal{R}}^d_{\le N}\) are uniform in \(N\), in contrast to our non-uniform bounds in \(N\) for \(\mathcal{R}^d_{\le N}\) given in 8 , the advantage of using Theorem \(\ref{thm:IW:upper}\) is twofold. First, it aligns with the set of canonical fractions \(\mathcal{R}^d_{\le N}\) that naturally arise in the circle method. Second, the set of canonical fractions allows us to relax the decay in condition 10 from exponential to polynomial, as in 7 . The latter was essential in establishing a multilinear variant of Weyl’s inequality in [19].

The penultimate remark can be quantified, leading to the second main result of the paper.

Theorem 2 demonstrates that applications of inequality 8 that would require its norm to be independent of the parameter \(N\) will not be possible. This is only possible with the Ionescu–Wainger fractions \(\widetilde{\mathcal{R}}^d_{\le N}\), as was proved by Tao [18].

Finally, let us emphasize that the method of proof of Theorem 2 can be adapted to show that condition [cond:B] imposed on our weights in Definition 1 is necessary; if dropped, one can show that Theorem 1 fails, see Remark 1 in Section 4.

1.3 Organization of the paper↩︎

In Section 2, we gather the necessary notation and some remarks on the weights from Definition 1. In Section 3 and Section 4, we prove Theorem 1 and Theorem 2, respectively. In Section 5, we introduce the concept of Rademacher–Menshov seminorms (see Definition 8) and establish a seminorm variant of Theorem 1 that generalizes [19]. Finally, in Section 6, we illustrate how our weighted Ionescu–Wainger theory works in practice, giving a relatively short and conceptually clean proof of Bourgain’s pointwise ergodic theorem for polynomial iterates, see Theorem 6. To this end, we prove its quantitative versions in Theorem 7. The latter provides the sharpest quantitative information on pointwise convergence for 2 .

2 Notation and simple tools↩︎

We now set up the notation that will be used throughout the article.

2.1 Basic notation↩︎

Let \(\mathbb{Z}_+ \mathrel{\vcenter{:}}= \{1,2,\dots\}\) be the set of positive integers, \({\mathbb{N}}\mathrel{\vcenter{:}}= \{0,1,2,\dots\}\) be the set of nonnegative integers, and \(\mathbb{R}_+ \mathrel{\vcenter{:}}= (0,\infty)\) be the set of positive real numbers. Given \(d \in \mathbb{Z}_+\), the sets \(\mathbb{Z}^d, \, {\mathbb{Q}}^d, \, \mathbb{R}^d\), and \({\mathbb{C}}^d\) have their standard meaning. The \(d\)-dimensional unit cube in \(\mathbb{R}^d\) centered at the origin is denoted by \({\boldsymbol{Q}}^d \mathrel{\vcenter{:}}= [-1/2,1/2]^d\).

If \(A\subseteq{\mathbb{C}}^d\), then its translations and dilations are defined by \(A+u\mathrel{\vcenter{:}}=\{a+u: a\in A\}\) for all \(u\in {\mathbb{C}}^d\) and \(vA\mathrel{\vcenter{:}}=\{va: a\in A\}\) for all \(v\in{\mathbb{C}}\), respectively.

For \(N \in \mathbb{R}_+\), we define \([N] \mathrel{\vcenter{:}}= \mathbb{Z}\cap (0,N]\). For \(N \in \mathbb{R}\) and \(A\subseteq \mathbb{R}\), we also define the sets \[\begin{align} A_{<N}&\mathrel{\vcenter{:}}= A\cap(-\infty, N), \quad \phantom{],} A_{>N}\mathrel{\vcenter{:}}= A\cap(N, \infty),\\ A_{\le N}&\mathrel{\vcenter{:}}= A\cap(-\infty, N], \quad \phantom{),} A_{\ge N}\mathrel{\vcenter{:}}= A\cap[N, \infty). \end{align}\]

Here and throughout, the \(d\)-dimensional torus \(\mathbb{T}^d \mathrel{\vcenter{:}}= \mathbb{R}^d / \mathbb{Z}^d\) is endowed with the \(\ell^\infty\) metric on \(\mathbb{T}^d\) induced by the norm \(| \xi |_\infty \mathrel{\vcenter{:}}= \sup_{i \in [d]} {\rm dist}(\xi_i, \mathbb{Z})\). Moreover, we will identify \(\mathbb{T}^d\) with \([0,1)^d\) or, up to a set of measure zero, with \({\boldsymbol{Q}}^d\), as will be clear from the context.

Finally, if \(A\) is a set, then its indicator function is denoted by \(\mathbb{1}_{{A}}\). When \(A\) is finite, its number of elements is denoted by \(|A|\). If \(S\) is a statement, then \(\mathbb{1}_{{S}}\) denotes its indicator, equal to \(1\) if \(S\) is true and \(0\) if \(S\) is false. In particular, we have \(\mathbb{1}_{{x\in A}} \equiv \mathbb{1}_{{A}}(x)\).

2.2 Asymptotic notation↩︎

Throughout the paper, \(C\in\mathbb{R}_+\) denotes an absolute constant whose value may change from line to line. For two quantities \(A,B\in\mathbb{R}_{\ge 0}\), we write \(A \lesssim B\) or \(B \gtrsim A\) if \(A\le CB\) for some \(C\in\mathbb{R}_+\). If \(A \lesssim B\lesssim A\), then we write \(A \simeq B\). We will use the symbols \(\lesssim_{\delta}\) or \(\simeq_{\delta}\) to emphasize that the implicit constant \(C\) depends on a parameter \(\delta\).

2.3 Function spaces↩︎

All vector spaces in this paper will be defined over the complex numbers \({\mathbb{C}}\). The triple \((X, \mathcal{B}(X), \mu)\) is a measure space \(X\) with \(\sigma\)-algebra \(\mathcal{B}(X)\) and \(\sigma\)-finite measure \(\mu\). The space of all measurable functions whose modulus is integrable with \(p\)-th power is denoted by \(L^p(X, \mu)\) for \(p\in(0, \infty)\), whereas \(L^{\infty}(X, \mu)\) denotes the space of all essentially bounded measurable functions. More generally, if \((B, \|\, \cdot \, \|_B)\) is a separable normed vector space, then \(L^{p}(X,\mu;B)\) denotes the space of all measurable functions \(F \colon X\to B\) (up to \(\mu\)-almost everywhere equivalence) such that \[\begin{align} \|F\|_{L^{p}(X, \mu;B)} \mathrel{\vcenter{:}}= \left\|\|F\|_B\right\|_{L^{p}(X, \mu)}<\infty. \end{align}\]

We shall abbreviate \(L^p(X, \mu)\) to \(L^p(X)\) and \(L^{p}(X,\mu;B)\) to \(L^{p}(X;B)\). If \(X\) is endowed with counting measure, then we further abbreviate \(L^p(X)\) to \(\ell^p(X)\) and \(L^p(X; B)\) to \(\ell^p(X; B)\).

For a continuous linear map \(T \colon B_1 \to B_2\) between two normed vector spaces \(B_1\) and \(B_2\), its operator norm will be denoted by \(\|T\|_{B_1 \to B_2}\). The space of all continuous linear mappings \(T \colon B_1 \to B_2\) will be denoted by \(L(B_1,B_2)\).

2.4 Fractions↩︎

For integers \(p, q\in\mathbb{Z}_+\), we write \(p\mid q\) if \(p\) divides \(q\) or, equivalently, \(q \in p \mathbb{Z}\).

For \(a = (a_1,\ldots, a_d) \in \mathbb{Z}^d\) and \(q\in\mathbb{Z}_+\), we denote by \({\rm gcd}(a,q)\) the largest positive integer \(n\in\mathbb{Z}_+\) that divides each of the integers \(q,a_1, \ldots, a_d\). Observe that any element of \({\mathbb{Q}}^d\) has a unique representation as \(a/q\) with \(a \in \mathbb{Z}^d\) and \(q\in \mathbb{Z}_{+}\) such that \({\rm gcd}(a,q)=1\).

Given \(N \in \mathbb{Z}_+\), define the \(1\)-periodic sets of canonical fractions by \[\begin{align} \label{IWeq:372} \mathcal{R}_{\le N}^d \mathrel{\vcenter{:}}= \big\{ a/q \in({\mathbb{Q}}/ \mathbb{Z})^d: q \in [N] \text{ and } {\rm gcd} (a, q)=1\big\}. \end{align}\tag{12}\] Here each element of \({\mathcal{R}}^d_{\le N}\) is represented by a real rational \(a/q \in {\mathbb{Q}}^d\) which we identify with the equivalence class \(a/q + \mathbb{Z}^d\). Similarly, for \(q \in \mathbb{Z}_+\), we define the set of \(q\)-quotients by \(\mathcal{Q}(q) \mathrel{\vcenter{:}}= (q^{-1} \mathbb{Z}/ \mathbb{Z})^d\) and the set of reduced fractions by \(\mathcal{R}(1) \mathrel{\vcenter{:}}= \mathcal{Q}(1)\) or, when \(q>1\), by \[{\mathcal{R}}(q) \mathrel{\vcenter{:}}= \{ a/q \in \mathcal{Q}(q) : {\rm gcd}(a,q) = 1 \}.\]

2.5 Convolutions↩︎

Let \((\mathbb{G}, +)\) be a locally compact abelian group (LCA group) equipped with a Haar measure \(\lambda_{\mathbb{G}}\). For two functions \(f, g\in L^1(\mathbb{G})\) we define their convolution by \[\begin{align} f*_{\mathbb{G}}g(x) \mathrel{\vcenter{:}}= f*g(x) \mathrel{\vcenter{:}}=\int_{\mathbb{G}}f(x-y)g(y)\,{\rm d}\lambda_{\mathbb{G}}(y). \end{align}\] For the sake of simplicity, we will always abbreviate \(*_{\mathbb{G}}\) to \(*\); this notation will always be clear from the context and will not cause any confusion. Since \(\mathbb{G}\) is abelian, we readily see that \(f*g=g*f\). We will be mainly working with \(\mathbb{G}=\mathbb{R}^d\) or \(\mathbb{G}=\mathbb{T}^d\) endowed with Lebesgue measure or \(\mathbb{G}=\mathbb{Z}^d\) endowed with counting measure.

2.6 Fourier transform↩︎

We shall write \(e(z)=e^{2\pi {\boldsymbol{i}} z}\) for every \(z\in{\mathbb{C}}\), where \({\boldsymbol{i}}^2=-1\). As before let \((\mathbb{G}, +)\) be an LCA group equipped with a Haar measure \(\lambda_{\mathbb{G}}\). It is well known (see for instance [21]) that every LCA group \(\mathbb{G}\) has a Pontryagin dual \(\hat{\mathbb{G}}= (\hat{\mathbb{G}},+)\), an LCA group with a Haar measure \(\lambda_{\hat{\mathbb{G}}}\) and a pairing, i.e. a continuous bihomomorphism \(\mathbb{G}\times \hat{\mathbb{G}}\ni (x,\xi) \mapsto \langle x , \xi\rangle\in\mathbb{T}\), such that the Fourier transform \(\mathcal{F}_{\mathbb{G}} \colon L^1(\mathbb{G}) \to L^\infty(\hat{\mathbb{G}})\) given by \[\mathcal{F}_{\mathbb{G}} f(\xi) \mathrel{\vcenter{:}}= \int_{\mathbb{G}} f(x) e(-\langle x , \xi\rangle)\,{\rm d}\lambda_{\mathbb{G}}(x), \qquad \xi\in\hat{\mathbb{G}},\] extends to a unitary map from \(L^2(\mathbb{G})\) to \(L^2(\hat{\mathbb{G}})\); in particular, we have Plancherel’s identity \[\|\mathcal{F}_\mathbb{G}f\|_{L^2(\hat{\mathbb{G}})}=\|f\|_{L^2(\mathbb{G})}, \qquad f\in L^2(\mathbb{G}).\] Moreover, the inverse Fourier transform \(\mathcal{F}_\mathbb{G}^{-1} \colon L^2(\hat{\mathbb{G}}) \to L^2(\mathbb{G})\) is given by the formula \[\mathcal{F}_\mathbb{G}^{-1} f(x) = \int_{\hat{\mathbb{G}}} f(\xi) e(\langle x , \xi\rangle)\,{\rm d}\lambda_{\hat{\mathbb{G}}}(\xi), \qquad x\in\mathbb{G},\] whenever \(f\in L^1(\hat{\mathbb{G}}) \cap L^2(\hat{\mathbb{G}})\).

We will mainly work with concrete pairs \((\mathbb{G},\hat{\mathbb{G}})\) of Pontryagin dual LCA groups:

• If \(\mathbb{G}= \mathbb{R}^d\) with Lebesgue measure \(\lambda_{\mathbb{R}^d} = {\rm d}x\), then \(\hat{\mathbb{G}}= \mathbb{R}^d\) with Lebesgue measure \(\lambda_{\mathbb{R}^d} = {\rm d}\xi\) is a Pontryagin dual, with pairing \(\langle x , \xi\rangle \mathrel{\vcenter{:}}= x \cdot \xi\). Also, for any \(f \in L^1(\mathbb{R}^d)\), \[\begin{align} \mathcal{F}_{\mathbb{R}^d} f(\xi) \mathrel{\vcenter{:}}= \int_{\mathbb{R}^d} f(x) e(-x\cdot\xi) \,{\rm d}x, \qquad \xi\in\mathbb{R}^d. \end{align}\]

• If \(\mathbb{G}= \mathbb{Z}^d\) with counting measure \(\lambda_{\mathbb{Z}^d}\), then \(\hat{\mathbb{G}}= \mathbb{T}^d\) with Lebesgue measure \(\lambda_{\mathbb{T}^d} = {\rm d}\xi\) is a Pontryagin dual, with pairing \(\langle x , \xi\rangle \mathrel{\vcenter{:}}= x \cdot \xi\). Also, for any \(f \in \ell^1(\mathbb{Z}^d)\), \[\begin{align} \mathcal{F}_{\mathbb{Z}^d}f(\xi) \mathrel{\vcenter{:}}= \sum_{x \in \mathbb{Z}^d} f(x) e(-x\cdot\xi), \qquad \xi\in \mathbb{T}^d. \end{align}\]

• If \(\mathbb{G}= (\mathbb{Z}/q\mathbb{Z})^d\) for some \(q \in \mathbb{Z}_+\) with normalized counting measure \(\lambda_{(\mathbb{Z}/q\mathbb{Z})^d}\), then \(\hat{\mathbb{G}} = (q^{-1}\mathbb{Z}/\mathbb{Z})^d\) with counting measure \(\lambda_{(q^{-1}\mathbb{Z}/\mathbb{Z})^d}\) is a Pontryagin dual, with pairing \(\langle x , \xi\rangle \mathrel{\vcenter{:}}= x \cdot \xi\). Also, for any \(f \in L^1((\mathbb{Z}/q\mathbb{Z})^d)\), \[\begin{align} \mathcal{F}_{(\mathbb{Z}/q\mathbb{Z})^d}f(\xi) \mathrel{\vcenter{:}}= q^{-d}\sum_{x \in (\mathbb{Z}/q\mathbb{Z})^d} f(x) e(-x\cdot\xi), \qquad \xi\in (q^{-1}\mathbb{Z}/\mathbb{Z})^d. \end{align}\]

2.7 Multiplier operators↩︎

For \(\mathfrak m \in L^\infty(\hat{\mathbb{G}})\), we define the multiplier operator by \[\begin{align} T_{\mathbb{G}}[\mathfrak m]f(x) \mathrel{\vcenter{:}}= \int_{\hat{\mathbb{G}}} \mathfrak m(\xi) \, \mathcal{F}_{\mathbb{G}}f(\xi) \, e(x \cdot \xi) \, {\rm d}\lambda_{\hat{\mathbb{G}}}(\xi), \qquad x\in \mathbb{G}, \end{align}\] where \(f \colon \mathbb{G}\to {\mathbb{C}}\) is any function for which this formula makes sense.

2.8 Multifrequency extensions↩︎

Let \(\mathfrak m \in L^\infty(\hat{\mathbb{G}})\), a finite set \(\Sigma \subseteq \hat{\mathbb{G}}\), and \(G \colon \hat{\mathbb{G}}\to \mathbb{C}\) be given. We define a weighted multifrequency extension of \(\mathfrak m\) with respect to \(\Sigma\) by setting \[\begin{align} \label{eq:1} \big\llbracket G \mid \mathfrak m\big\rrbracket_{\Sigma}(\xi)\mathrel{\vcenter{:}}=\sum_{\sigma\in\Sigma}G(\sigma)\mathfrak m(\xi-\sigma), \qquad \xi\in \hat{\mathbb{G}}. \end{align}\tag{13}\] If \(G\equiv 1\), then \(\llbracket 1 \mid \mathfrak m\rrbracket_{\Sigma}\) is the multifrequency extension of \(\mathfrak m\) with respect to \(\Sigma\). If \(\Sigma=\{\sigma\}\), then we write \(\llbracket G \mid \mathfrak m\rrbracket_{\Sigma}=\llbracket G \mid \mathfrak m\rrbracket_{\sigma}\). If \(\Sigma\) is the singleton of the neutral element of \(\hat{\mathbb{G}}\), then we simply have \(\llbracket 1 \mid \mathfrak m\rrbracket_{\Sigma}=\mathfrak m\).

2.9 Gaussian weights↩︎

Let \(w\colon \mathbb{Z}^k \to \mathbb{C}\) be a weight that satisfies for some \(C_w \in \mathbb{R}_+\) the following \(\ell^1\) bound \[\label{eq:w-L1} \sup_{q\in\mathbb{Z}_+}q^{-k} \sum_{r\in[q]^k} |w(r)| \le C_w.\tag{14}\] Here, we shall show that, for any function \(\Phi \colon \mathbb{Z}^k\to \mathbb{Z}^d\), a generalized Gaussian sum \[\begin{align} G_w(a/q)\mathrel{\vcenter{:}}= q^{-k}\sum_{r\in[q]^k}w(r) \, e(\Phi(r) \cdot a/q), \qquad a/q\in({\mathbb{Q}}/\mathbb{Z})^d, \end{align}\] satisfies condition [cond:B]. Indeed, fix \(d, k \in \mathbb{Z}_+\) and \(\Phi\colon \mathbb{Z}^k\to\mathbb{Z}^d\). We show that \(G_w\) satisfies condition [cond:B] with \(U_G = C_w\), that is, \[\label{eq:Gauss-IW} \sup_{q \in \mathbb{Z}_+} q^{-d} \sum_{m \in (\mathbb{Z}/ q \mathbb{Z})^d} \Big| \sum_{a \in [q]^d} G_w(a/q)\, e(m \cdot a/q) \Big| \le C_w.\tag{15}\]

Fix \(q \in \mathbb{Z}_+\). For each \(m\in{(\mathbb{Z}/ q \mathbb{Z})^d}\), we have \[\mathcal{F}^{-1}_{(\mathbb{Z}/ q \mathbb{Z})^d} (G_w)(m) = \sum_{a \in [q]^d} G_w(a/q)\, e(m \cdot a/q).\] Substituting the definition of \(G_w\), interchanging sums, and using orthogonality, we obtain \[\begin{align} \mathcal{F}^{-1}_{(\mathbb{Z}/ q \mathbb{Z})^d} (G_w)(m) = q^{-k} \sum_{r\in[q]^k} w(r) \sum_{a \in [q]^d} e\big( (\Phi(r) + m) \cdot a / q \big)=q^{-k+d} \sum_{\substack{r\in[q]^k\\ \Phi(r) + m\in q\mathbb{Z}^d}} w(r). \end{align}\] By taking absolute values, summing over \(m\), and interchanging summation order, we have \[\begin{align} \sum_{m \in (\mathbb{Z}/ q \mathbb{Z})^d} |\mathcal{F}^{-1}_{(\mathbb{Z}/ q \mathbb{Z})^d} (G_w)(m)| \leq q^{-k+d} \sum_{r\in[q]^k} |w(r)|, \end{align}\] since for each \(r\in[q]^k\) exactly one \(m \in (\mathbb{Z}/ q \mathbb{Z})^d\) satisfies \(\Phi(r) + m \in q\mathbb{Z}^d\). Then 14 gives \[\| \mathcal{F}^{-1}_{(\mathbb{Z}/ q \mathbb{Z})^d} (G_w) \|_{L^1({(\mathbb{Z}/ q \mathbb{Z})^d})} = q^{-d} \sum_{m \in (\mathbb{Z}/ q \mathbb{Z})^d} |\mathcal{F}^{-1}_{(\mathbb{Z}/ q \mathbb{Z})^d} (G_w)(m)| \le q^{-k} \sum_{r\in[q]^k} |w(r)| \leq C_w,\] as desired. In particular, if \(w\) is nonnegative and normalized by \(\sum_{r\in[q]^k} w(r) = q^k\), then \(\mathcal{F}^{-1}_{(\mathbb{Z}/ q \mathbb{Z})^d} (G_w) \ge 0\) and the above inequality becomes an equality with \(C_w = 1\).

3 Proof of Theorem 1: the upper bound↩︎

In this section, we prove Theorem 1. Although its content shares many similarities with [15] or [19], there are some subtle differences that may cause difficulties, so for convenience we include detailed explanations to ease the exposition. We begin with the formulation of the sampling principle of Magyar, Stein, and Wainger [22].

3.1 Simple reductions↩︎

We begin the proof of Theorem 1 with a list of reductions.

1. Since the case \(r=1\) follows by Plancherel’s theorem, we may assume that \(r \in {\mathbb{N}}_{\geq 2}\).

2. By duality and interpolation, it suffices to prove 8 for \(p = 2r\).

3. By replacing \(\mathfrak m\) with \(\tilde{\mathfrak m} \colon \mathbb{R}^d \to L(H_1\oplus H_2, H_1\oplus H_2)\) defined by \[\tilde{\mathfrak m}(x) (h_1, h_2) \mathrel{\vcenter{:}}= (0,\mathfrak m(x)(h_1)),\] we may assume that \(H_1=H_2=H\).

4. By standard limiting arguments, we may assume that \(H\) is finite-dimensional.

5. We may assume that \(\mathbf{A}_{2r} \in (0,\infty)\) and then normalize \(\mathfrak m\) so that \(\mathbf{A}_{2r} = 1\).

6. Since Theorem 3 applied with \(q=1\) gives 8 with \(N^{C_r / \log \log N} \mathbf{C}_{G}(\mathcal{R}^d_{\leq N})\) replaced by \(N^{d+1}\), we may assume that \(N \geq N_0\) for a fixed large integer \(N_0\in\mathbb{Z}_+\).

After all these reductions, our aim is to prove 8 with \(p=2r\), which reads as follows \[\big\| T_{\mathbb{Z}^d} \big[ \llbracket G \mid \mathfrak m\rrbracket_{\mathcal{R}^d_{\le N}}\big] f \big\|_{\ell^{2r}(\mathbb{Z}^d; H)} \lesssim_{d,r} N^{{\boldsymbol{C}}_{\rm IW}(r) / \log \log N} \mathbf{C}_{G}(\mathcal{R}^d_{\leq N}) \|f\|_{\ell^{2r}(\mathbb{Z}^d; H)}.\]

3.2 Uniqueness property↩︎

We recall the following definition, which was essential in [11].

Using Definition 2, we now formulate an important abstract orthogonality principle.

Our next goal is to partition the set of canonical fractions 12 into a controlled number of pieces to which Proposition 1 can be applied efficiently. Here our approach significantly deviates from [11] and [15]. In these two papers an essential role was played by the Ionescu–Wainger rationals whose denominators were factored into two parts, one being a relatively large product of powers of small prime divisors, and the other being a product of a small number of powers of large prime divisors. In contrast to [11] and [15], instead of working with the Ionescu–Wainger rationals, we work directly with the set of canonical fractions 12 and partition it using the new concept of \(N\)-lifted composites introduced in [19].

We recall the concept of property \(\mathcal{O}_N\) from [19], a variant of property \(\mathcal{O}\) from [11].

Let us enumerate the set of all prime numbers \(\mathbb{P} = \{p_m : m \in \mathbb{Z}_+ \}\) in such a way that \(p_{m'} > p_{m}\) holds if \(m' > m\). The following lemma explains the restriction \(K \in [K_N]\).

From now on, we assume that \(N_0\in\mathbb{Z}_+\) is so large that, in particular, Lemma 1 and the upper bound \(|\mathbb{P} \cap {\mathbb{N}}_{\leq N}| \leq \lfloor \frac{2 N}{\log N} \rfloor\) from the prime number theorem are applicable if \(N \geq N_0\).

3.3 Partitioning of the set of canonical fractions↩︎

For \(N\in {\mathbb{N}}_{\geq N_0}\), let \(I_N \mathrel{\vcenter{:}}= \lfloor \frac{2 N}{\log N} \rfloor\). As observed before, for every \(a/q \in \mathcal{R}^d_{\leq N}\), the corresponding composite \(Q_q\) consists of at most \(K_N\) factors of the form \(p^{\lfloor \log_p N \rfloor}\), where each \(p\) is one of the first \(I_N\) prime numbers. This motivates the following counting lemma that originates in [12].

Proceeding in two steps, we partition \(\mathcal{R}^d_{\le N}\) into pieces, each having property \(\mathcal{O}_N\).

3.3.0.1 Step 1

By Lemma 1, we partition \(\mathcal{R}^d_{\le N}\) into disjoint subsets \(\mathcal{R}^d_{N,0} \mathrel{\vcenter{:}}= \mathcal{R}(1)\) and \(\mathcal{R}^d_{N,K}\) for \(K \in [K_N]\), consisting of all fractions \(a/q \in \mathcal{R}^d_{\le N}\) whose denominators \(q\) have exactly \(K\) prime divisors. The multiplier operator corresponding to \(\mathcal{R}^d_{N,0}\) can be readily handled by using Theorem 3. Hence, from now on, we focus only on \(\mathcal{R}^d_{N,K}\) with \(K \in [K_N]\).

3.3.0.2 Step 2

Fix \(K \in [K_N]\). We apply Lemma 2 with \[V \mathrel{\vcenter{:}}= \{ p^{\lfloor \log_p N \rfloor} : p \in \mathbb{P} \cap {\mathbb{N}}_{\leq N} \} = \{ Q_{p} : p \in \mathbb{P} \cap {\mathbb{N}}_{\leq N} \},\] which produces surjective functions \(g_1,\ldots, g_J \colon V\to [K]\) with \(J \in [J_N]\). For every \(j \in[J]\) and \(k \in [K]\), we define \(S_k^j \mathrel{\vcenter{:}}= g_j^{-1}(\{k\}) = \{ v \in V : g_j(v) = k \}\). Next, we iteratively define \[\begin{align} \mathcal{O}(g_1) & \mathrel{\vcenter{:}}= S_1^1\cdots S_K^1, \\ \mathcal{O}(g_2) & \mathrel{\vcenter{:}}= S_1^2\cdots S_K^2 \setminus \mathcal{O}(g_1), \\ & \;\;\vdots \\ \mathcal{O}(g_J) & \mathrel{\vcenter{:}}= S_1^J\cdots S_K^J \setminus \big(\mathcal{O}(g_1) \cup \dots \cup \mathcal{O}(g_{J-1})\big). \end{align}\] Finally, for each \(j \in [J]\), we set \[\begin{align} \label{IWeq:8} \mathcal{R}^d_{N,K,j} \mathrel{\vcenter{:}}= \big\{ a / q \in \mathcal{R}^d_{N,K}: Q_q\in \mathcal{O}(g_j) \big\}. \end{align}\tag{16}\] The sets \(\mathcal{R}^d_{N,K,j}\) are disjoint and the corresponding sets \(\mathcal{O}(g_j)\) have property \(\mathcal{O}_N\). Moreover, \(\mathcal{R}^d_{N,K}= \mathcal{R}^d_{N,K,1} \cup \dots \cup \mathcal{R}^d_{N,K,J}\). Indeed, if \(a/q \in \mathcal{R}^d_{N,K}\), then \(q\) has exactly \(K\) prime divisors, say \(p_1, \ldots, p_K\in\mathbb{P}\), and \(Q_q=Q_{p_1}\cdots Q_{p_K}\). By Lemma 2, there exists \(j\in[J]\) such that \(|g_j(\{Q_{p_1}, \dots, Q_{p_K}\})|=K\). Taking the smallest index \(j\) with this property, we obtain \(Q_q=Q_{p_1}\cdots Q_{p_K}\in \mathcal{O}(g_j)\) and consequently \(a/q \in \mathcal{R}^d_{N,K, j}\), as desired.

The product structure in \(S_1^j\cdots S_K^j\) is crucial for employing denominator and numerator orthogonality arguments. An alternative way to reveal such a product structure is to apply the random partition technique appearing in the proof of [18].

To complete the proof, it suffices to prove inequality 8 with \(p=2r\) and the multiplier \(\llbracket G \mid \mathfrak m\rrbracket_{\mathcal{R}^d_{\le N}}\) replaced by \(\llbracket G \mid \mathfrak m\rrbracket_{\mathcal{R}^d_{N,K,j}}\) for every \(K \in [K_N]\) and \(j\in [J]\), with \(\mathcal{R}^d_{N,K,j}\) defined in 16 . For this purpose, additional notation and terminology need to be introduced.

3.4 \(\mathbb{P}\)-irreducibility and factorization↩︎

Let us first recall the concept of \(\mathbb{P}\)-irreducible fractions from [19], which played a critical role in the verification of condition ?? .

We note that \(\mathcal{I}(q)\supseteq \mathcal{R}(q)\). Moreover, \(\mathcal{I}(q)\) can be expressed as a disjoint union of \(\mathcal{R}(\tilde{q})\), where \(\tilde{q}\) runs over all positive divisors of \(q\) having exactly the same prime divisors as \(q\).

For a finite set \(\Lambda \subset \mathbb{Z}_+\), define \[\mathcal{R}(\Lambda) \mathrel{\vcenter{:}}= \bigcup_{q \in \Lambda} \mathcal{R}(q) \quad \text{and} \quad \mathcal{I}(\Lambda) \mathrel{\vcenter{:}}= \bigcup_{q \in \Lambda} \mathcal{I}(q).\] Obviously, \(\mathcal{R}(\Lambda)\subseteq \mathcal{I}(\Lambda)\) and the set of canonical fractions 12 can be written as \(\mathcal{R}([N])\) using this definition. An important feature of \(\mathcal{R}(\Lambda)\), which was used in [11] and [15], is the following factorization property. If \({\rm gcd}(q, q') = 1\) for every pair \((q,q') \in \Lambda \times \Lambda'\), then \[\mathcal{R}(\Lambda \Lambda') = \mathcal{R}(\Lambda) \oplus \mathcal{R}(\Lambda').\] The direct sum \(\oplus\) indicates that each element \(u\) in the sumset \({\mathcal{R}}(\Lambda) + {\mathcal{R}}(\Lambda')\) has a unique representation \(u = a/q + a'/q' \pmod{\mathbb{Z}^d}\). This follows from the Chinese remainder theorem.

The following definition allows us to formulate the factorization property for \(\mathcal{I}(\Lambda)\).

Fix \(K \in [K_N]\) and \(j\in [J]\). Referring to 16 , we observe that \[\llbracket G \mid \mathfrak m\rrbracket_{\mathcal{R}^d_{N,K,j}} = \llbracket G^\circ \mid \mathfrak m\rrbracket_{\mathcal{I}(S_1 \cdots S_K)}\] with \(S_k \mathrel{\vcenter{:}}= g_j[\{k\}]^{-1}\) for all \(k \in [K]\) and \(G^\circ \mathrel{\vcenter{:}}= G \cdot \mathbb{1}_{{N,K,j}}\), where \(\mathbb{1}_{{N,K,j}}(a/q) \mathrel{\vcenter{:}}= 1\) if \(a/q\) is \(\mathbb{P}\)-irreducible and upon reduction belongs to \(\mathcal{R}^d_{N,K,j}\), and \(\mathbb{1}_{{N,K,j}}(a/q) \mathrel{\vcenter{:}}= 0\) otherwise. The sets \(S_k\) form a partition of \(V\) determined by the surjection \(g_j\). Moreover, the \({\mathbb{P}}\)-irreducible fractions \({\mathcal{I}}(q)\) are pairwise disjoint sets as \(q\) varies over the product set \(S_1 \cdots S_K\).

3.5 Denominators↩︎

We use Lemma 3 repeatedly to exploit orthogonality between the denominators. Denote \(f^\circ_{\mathcal{I}} \mathrel{\vcenter{:}}= T_{\mathbb{Z}^d} \big[\llbracket G^\circ \mid \mathfrak m\rrbracket_{\mathcal{I}}\big]f\) for every \(\mathcal{I} \subseteq \mathcal{I}(S_1 \cdots S_K)\). We have \[\begin{align} \label{denom-0} \big\| T_{\mathbb{Z}^d} \big[ \llbracket G \mid \mathfrak m\rrbracket_{\mathcal{R}^d_{N,K,j}} \big] f \big\|_{\ell^{2r}(\mathbb{Z}^d; H)}^{2r} = \sum_{x \in \mathbb{Z}^d} \big\| f^\circ_{\mathcal{I}(S_1 \cdots S_K)} (x) \big\|_{H}^{2r}. \end{align}\tag{17}\]

For \(L \subseteq [K]\), set \({\mathcal{S}}_L \mathrel{\vcenter{:}}= \prod_{l \in L} S_l\) with the convention \({\mathcal{S}}_{\emptyset} \mathrel{\vcenter{:}}= \{1\}\). Moreover, for \(q \in {\mathcal{S}}_L\), we let \(F_{q, L} \mathrel{\vcenter{:}}= f^\circ_{\mathcal{I}(q) \oplus \mathcal{I}({\mathcal{S}}_{[K] \setminus L})}\). Now, for each \(k \in [K]\), we shall show the following inequality \[\begin{align} \label{denom} \sum_{x \in \mathbb{Z}^d} \Big( \sum_{q \in {\mathcal{S}}_{[k-1]}} \big \| F_{q, [k-1]} (x) \big \|_{H}^{2} \Big)^r \leq C_r \sum_{x \in \mathbb{Z}^d} \Big( \sum_{q \in {\mathcal{S}}_{[k]}} \big \| F_{q, [k]}(x) \big \|_{H}^{2} \Big)^r \end{align}\tag{18}\] with \(C_r = {\boldsymbol{C}}_{\rm UP}(r)\) using Proposition 1. After \(K\) iterations of 18 , we shall obtain \[\begin{align} \sum_{x \in \mathbb{Z}^d} \Big( \sum_{q \in {\mathcal{S}}_{\emptyset}} \big \| F_{q,\emptyset}(x) \big \|_{H}^{2} \Big)^r \leq C_r^{K} \sum_{x \in \mathbb{Z}^d} \Big( \sum_{q \in {\mathcal{S}}_{[K]}} \big \| F_{q, [K]}(x) \big \|_{H}^{2} \Big)^r \end{align}\] or, equivalently, \[\begin{align} \label{denom-II} \sum_{x \in \mathbb{Z}^d} \big\| f^\circ_{\mathcal{I}(S_1 \cdots S_K)} (x) \big\|_{H}^{2r} \le C_r^{K} \sum_{x \in \mathbb{Z}^d} \Big(\sum_{q \in S_{1} \cdots S_{K}} \big\| f^\circ_{\mathcal{I}(q)} (x) \big\|_{H}^{2}\Big)^r. \end{align}\tag{19}\] The constant \(C_r^{K/2r}\) is acceptable because \(K \leq K_N = \lfloor \frac{5 \log N}{\log \log N} \rfloor\).

Turning to the proof of 18 , we rewrite the left-hand side as \[\sum_{q_1, \dots, q_r \in {\mathcal{S}}_{[k-1]}} \sum_{x \in \mathbb{Z}^d} \prod_{i \in [r]} \Big \| \sum_{q^* \in S_{k}} F_{q_i q^*, [k]}(x) \Big\|_{H}^{2}.\] Fix \(q_1, \dots, q_r \in {\mathcal{S}}_{[k-1]}\). We claim that, for every \(2r\)-tuple \((q^*_{1,1}, q^*_{1,2}, \dots, q^*_{r,1}, q^*_{r,2}) \in S_{k}^{2r}\) with the uniqueness property, the Fourier transform of the function \[\prod_{i \in [r]} \langle F_{q_i q^*_{i,1}, [k]}, F_{q_i q^*_{i,2}, [k]} \rangle_{H}\] vanishes in a neighborhood of the origin so that ?? holds. Indeed, since \(H\) is separable, the Fourier transform of this function is a finite sum of functions, each supported on a set \[2r \varepsilon {\boldsymbol{Q}}^d + (v_{1,1} - v_{1,2}) + \cdots + (v_{r,1} - v_{r,2}),\] where \(v_{i,1}, v_{i,2} \in \mathcal{I}(S_{[K]})\) are fractions with denominators divisible by \(q^*_{i,1}, q^*_{i,2}\) respectively for each \(i\in[r]\). Moreover, these denominators, in reduced form, are not greater than \(N\). We rewrite the above set as \(2r \varepsilon {\boldsymbol{Q}}^d + v\) and observe that \(v \notin \mathbb{Z}^d\) because, due to the uniqueness property, property \(\mathcal{O}_N\), and \(\mathbb{P}\)-irreducibility, at least one of the factors \(q^*_{1,1}, q^*_{1,2}, \dots, q^*_{r,1}, q^*_{r,2}\) cannot be completely eliminated from the denominator of \(v\). Thus, for some \(i \in [d]\), the distance between \(\mathbb{Z}\) and the \(i\)-th coordinate of \(v\) is at least \(N^{-2r}\). By \(\varepsilon \leq (4r N^{2r})^{-1}\), this implies that \(\mathbb{Z}^d \cap (2r \varepsilon {\boldsymbol{Q}}^d +v) = \emptyset\). Since condition ?? is satisfied, an application of Proposition 1 completes the proof of inequality 18 .

3.6 Numerators↩︎

Next, we exploit orthogonality between the numerators to bound the right-hand side of 19 . As before, we use Lemma 3 repeatedly. Our strategy is to split the sums in question into diagonal and off-diagonal terms by using the following result.

For any partition \(L \cup L' \cup L'' = [K]\), we define \[{\mathcal{F}}_{L,L',L''} \mathrel{\vcenter{:}}= \sum_{q \in {\mathcal{S}}_L} \Big( \sum_{q' \in {\mathcal{S}}_{L'}} \sum_{q'' \in {\mathcal{S}}_{L''}} \sum_{u'' \in \mathcal{I}(q'')} \big \| f^\circ_{\mathcal{I}(q) \oplus \mathcal{I}(q')+u''} \big \|_{H}^{2} \Big)^r.\] Then the sum on the right-hand side of 19 can be rewritten as follows \[\begin{align} \label{denom-III} \sum_{x \in \mathbb{Z}^d} \Big(\sum_{q \in S_{1} \cdots S_{K}} \big\| f^\circ_{\mathcal{I}(q)} (x) \big\|_{H}^{2}\Bigr)^r = \| {\mathcal{F}}_{\emptyset, [K], \emptyset} \|_{\ell^1(\mathbb{Z}^d)}. \end{align}\tag{20}\] Assume that \(l \in L'\) and set \(L_+ \mathrel{\vcenter{:}}= L \cup \{l\}, \, L'_- \mathrel{\vcenter{:}}= L' \setminus \{l\}, \, L''_+ \mathrel{\vcenter{:}}= L'' \cup \{l\}\). We shall prove \[\begin{align} \label{numer} \| {\mathcal{F}}_{L, L',L''} \|_{\ell^1(\mathbb{Z}^d)} \leq C_r \big( \| {\mathcal{F}}_{L_+, L'_-,L''} \|_{\ell^1(\mathbb{Z}^d)} + \| {\mathcal{F}}_{L, L'_-,L''_+}\|_{\ell^1(\mathbb{Z}^d)} \big) \end{align}\tag{21}\] with \(C_r = {\boldsymbol{C}}_{\rm diag}(r)(1+{\boldsymbol{C}}_{\rm UP}(r))\) using Lemma 4 and Proposition 1. This gives \[\begin{align} \label{IWeq:9} \| {\mathcal{F}}_{\emptyset, [K], \emptyset} \|_{\ell^1(\mathbb{Z}^d)} \leq C_r^K \sum_{L \subseteq [K]} \| {\mathcal{F}}_{L,\emptyset,[K] \setminus L} \|_{\ell^1(\mathbb{Z}^d)} \end{align}\tag{22}\] after \(K\) iterations of 21 . The implied constant in 22 is acceptable because \(K \leq K_N\).

Regarding 21 , by Lemma 3 we can expand \(\| {\mathcal{F}}_{L, L',L''} \|_{\ell^1(\mathbb{Z}^d)}\) as follows \[\sum_{x \in \mathbb{Z}^d} \sum_{q \in {\mathcal{S}}_{L}} \Big( \sum_{q^* \in S_l} \sum_{q' \in {\mathcal{S}}_{L'_-}} \sum_{q'' \in {\mathcal{S}}_{L''}} \sum_{u'' \in\mathcal{I}(q'')} \big \| f^\circ_{\mathcal{I}(q) \oplus \mathcal{I}(q^*) \oplus \mathcal{I}(q') +u''}(x) \big\|_{H}^{2} \Big)^r.\] Applying Lemma 4 to the sum over \(S_l\), we see that \(\| {\mathcal{F}}_{L, L',L''} \|_{\ell^1(\mathbb{Z}^d)}\) is controlled by the diagonal sum, which is equal to \(\| {\mathcal{F}}_{L_+, L'_-,L''} \|_{\ell^1(\mathbb{Z}^d)}\), and the off-diagonal sum corresponding to distinct \(q^*_1,\ldots, q^*_r \in S_l\). To control the latter by \(\| {\mathcal{F}}_{L, L'_-,L''_+}\|_{\ell^1(\mathbb{Z}^d)}\), we expand the \(r\)-fold product and, for fixed \(q \in {\mathcal{S}}_L, \, q'_i \in {\mathcal{S}}_{L'_-}, \, q''_i \in {\mathcal{S}}_{L''}, \, u''_i \in \mathcal{I}(q_i'')\) and distinct \(q^*_i \in S_l\), we set \(F_{u^*_i} \mathrel{\vcenter{:}}= f^\circ_{\mathcal{I}(q) \oplus \mathcal{I}(q_i') + u^*_i +u_i''}\), and we claim that Proposition 1 implies \[\begin{align} \label{numer2} \sum_{x \in \mathbb{Z}^d} \prod_{i \in [r]} \Big \| \sum_{u^*_i \in\mathcal{I}(q^*_i)} F_{u^*_i}(x) \Big\|_{H}^{2} \leq {\boldsymbol{C}}_{\rm UP}(r) \sum_{x \in \mathbb{Z}^d} \prod_{i \in [r]} \sum_{u^*_i \in\mathcal{I}(q^*_i)} \|F_{u^*_i}(x)\|_{H}^{2}. \end{align}\tag{23}\] Having established 23 , we can complete the proof of 21 by replacing the off-diagonal sum with the full sum and then combining the appropriate sums into \(r\)-th powers.

Turning to the proof of 23 , we claim that, for every \(2r\)-tuple \((u^*_{1,1}, u^*_{1,2}, \dots, u^*_{r,1}, u^*_{r,2}) \in \mathcal{I}(q^*_{1})^2 \times \cdots \times \mathcal{I}(q^*_{r})^2\) with the uniqueness property, the Fourier transform of the function \[\prod_{i \in [r]} \langle F_{u^*_{i,1}}, F_{u^*_{i,2}} \rangle_{H}\] vanishes in a neighborhood of the origin so that ?? holds. Indeed, since \(H\) is separable, the Fourier transform of this function is a finite sum of functions, each supported on a set \[2r \varepsilon {\boldsymbol{Q}}^d + (u^*_{1,1} - u^*_{1,2} + v_{1,1} - v_{1,2}) + \dots + (u^*_{r,1} - u^*_{r,2} + v_{r,1} - v_{r,2}),\] where \(u^*_{i,1} + v_{i,1}, \, u^*_{i,2} + v_{i,2}\) are fractions that, in reduced form, have denominators at most \(N\). We rewrite the above set as \(2r \varepsilon {\boldsymbol{Q}}^d + v\) and observe that \(v \notin \mathbb{Z}^d\) because, due to the uniqueness property, property \(\mathcal{O}_N\), and \(\mathbb{P}\)-irreducibility, for at least one fraction \(u^*_{i,1} - u^*_{i,2} + v_{i,1} - v_{i,2}\) the corresponding factor \(q^*_i \in S_l\) cannot be completely eliminated from the denominator of \(u^*_{i,1} - u^*_{i,2} + v_{i,1} - v_{i,2}\), hence the same is true for \(v\) by property \(\mathcal{O}_N\) because \(q^*_i\) are distinct. Thus, for some \(i \in [d]\), the distance between \(\mathbb{Z}\) and the \(i\)-th coordinate of \(v\) is at least \(N^{-2r}\). By \(\varepsilon \leq (4r N^{2r})^{-1}\), this implies \(\mathbb{Z}^d \cap (2r \varepsilon {\boldsymbol{Q}}^d +v) = \emptyset\). Since condition ?? is satisfied, an application of Proposition 1 completes the proof of 23 .

3.7 Square function estimates↩︎

Gathering 17 , 19 , 20 , and 22 together, we see that the verification of 8 has been reduced to proving \[\begin{align} \sum_{L \subseteq [K]} \| {\mathcal{F}}_{L,\emptyset,[K] \setminus L} \|_{\ell^1(\mathbb{Z}^d)} \lesssim_{r,d} N^{2rC / \log \log N} \mathbf{C}_{G}(\mathcal{R}^d_{\le N})^{2r} \|f\|_{\ell^{2r}(\mathbb{Z}^{d};H)}^{2r}. \end{align}\] Since there are \(2^K\) subsets \(L \subseteq [K]\), we can deal with each term in the sum above separately at the expense of introducing a factor \(C_r 2^{C_r \log N / \log \log N}\) which is acceptable.

We fix \(L \subseteq [K]\) and set \(\mathcal{I}' \mathrel{\vcenter{:}}= \mathcal{I}(\mathcal{S}_{[K] \setminus L}) \cap \mathcal{R}^d_{\le N}\). Since \(\mathbb{1}_{{N,K,j}}(u+u') = 0\) for \(u \in \mathcal{I}(q)\) and \(q \in S_L\) when \(u' \in \mathcal{I}(\mathcal{S}_{[K] \setminus L}) \setminus \mathcal{R}^d_{\le N}\), we are reduced to showing the inequality \[\begin{align} \label{IWeq:10} \sum_{x \in \mathbb{Z}^d} \sum_{q \in \mathcal{S}_L} \Big( \sum_{u' \in \mathcal{I}'} \big \| f^\circ_{\mathcal{I}(q) +u'}(x) \big\|_{H}^{2} \Big)^r \lesssim_{r,d} N^{2r C /\log \log N} \mathbf{C}_{G}(\mathcal{R}^d_{\le N})^{2r} \|f\|_{\ell^{2r}(\mathbb{Z}^{d};H)}^{2r}, \end{align}\tag{24}\] which we view as a linear operator norm bound from \(\ell^{2r}(\mathbb{Z}^{d};H)\) to \(\ell^{2r}(\mathbb{Z}^{d} \times \mathcal{S}_{L};\ell^{2}( \mathcal{I}' ;H))\). The reduction to 24 uses the fact that \(\mathcal{I}(q) \cap \mathcal{I}(q') = \emptyset\) holds for distinct \(q, q' \in \mathcal{S}_{[K]\setminus L}\).

The following lemma is the key improvement, resulting in a factor of \(1 / \log \log N\) in the exponent rather than the \(\log \log \log N / \log \log N\) factor used in [19].

Using Lemma 5, we partition \(\mathcal{I}(q) \cap \mathcal{R}^d_{\le N}\) into disjoint sets \(\mathcal{R}(b_1), \dots,\mathcal{R}(b_{M})\) for some \(M \in [M_N]\) and \(b_1, \dots, b_{M} \in \mathcal{B}_q\). Note that the set \(\mathcal{I}(q)\) appearing on the left-hand side of 24 can be replaced with \(\mathcal{I}(q) \cap \mathcal{R}^d_{\le N}\). Hence, denoting \(F_{\mathcal{I}}(x)\mathrel{\vcenter{:}}= \big( \sum_{u' \in \mathcal{I}'} \| f^\circ_{\mathcal{I}+u'}(x) \|_{H}^{2} \big)^r\) for \(\mathcal{I} \subseteq \mathcal{I}(q)\), we can bound the left-hand side of 24 as follows \[\begin{align} \sum_{x \in \mathbb{Z}^d} \sum_{q \in \mathcal{S}_L} F_{\mathcal{I}(q)}(x)\le M_N^{2r-1} \sum_{q \in \mathcal{S}_L}\sum_{b \in \mathcal{B}_q} \|F_{\mathcal{R}(b)}\|_{\ell^1(\mathbb{Z}^d)} \le M_N^{2r} \sum_{q \in \mathcal{S}_L}\sup_{b \in \mathcal{B}_q} \|F_{\mathcal{R}(b)}\|_{\ell^1(\mathbb{Z}^d)} \end{align}\] using the triangle inequality and Hölder’s inequality. Let \(f_{\mathcal{I}} \mathrel{\vcenter{:}}= T_{\mathbb{Z}^d}\big[\llbracket G \mid \mathfrak m\rrbracket_{\mathcal{I}}\big] f\) be a variant of \(f^\circ_{\mathcal{I}}\) without the factor \(\mathbb{1}_{{N,K,j}}\). By linearizing the supremum, we are reduced to proving \[\begin{align} \sum_{x \in \mathbb{Z}^d} \sum_{q \in {\mathcal{S}}_L} \Big( \sum_{u' \in \mathcal{I}'} \big \| f_{\mathcal{R}(b_q)+u'} (x) \big\|_{H}^{2} \Big)^r \lesssim_{r,d} N^{2r C / \log \log N} \mathbf{C}_{G}(\mathcal{R}^d_{\leq N})^{2r} \|f\|_{\ell^{2r}(\mathbb{Z}^{d};H)}^{2r} \end{align}\] uniformly in all possible choices of admissible divisors \(b_q \in \mathcal{B}_q\). Indeed, this follows because \(\mathbb{1}_{{N,K,j}}\) is constant on each set \(\mathcal{R}(b_q) + u'\) and vanishes on \(\mathcal{I}(q) \setminus \mathcal{R}^d_{\le N}\).

Now, for each fixed choice of admissible divisors \(b_q\) of \(q\in {\mathcal{S}}_L\), we shall obtain \[\begin{align} \label{IWeq:again} \sum_{x \in \mathbb{Z}^d} \sum_{q \in {\mathcal{S}}_L} \Big( \sum_{u' \in \mathcal{I}'} \big \| f_{\mathcal{R}(b_q)+u'} (x) \big\|_{H}^{2} \Big)^r \lesssim_{r,d} \sum_{x \in \mathbb{Z}^d} \sum_{q \in {\mathcal{S}}_L} \Big( \sum_{u' \in \mathcal{I}'} \big \| f^{\zeta_\varepsilon}_{\mathcal{R}(b_q)+u'} (x) \big\|_{H}^{2} \Big)^r, \end{align}\tag{27}\] where \(f_{\mathcal{I}}^{\zeta_\varepsilon} \mathrel{\vcenter{:}}= T_{\mathbb{Z}^d} \big[\llbracket G \mid {\zeta_\varepsilon}\rrbracket_{\mathcal{I}}\big] f\) is a variant of \(f_{\mathcal{I}}\) with \(\mathfrak m\) replaced by a bump multiplier \({\zeta_\varepsilon}\). We shall use the vector-valued Marcinkiewicz–Zygmund theorem. Indeed, once \(q \in {\mathcal{S}}_{L}\) is fixed, both sides of 27 are \(2r\)-th powers of norms of some \(\ell^{2}(\mathcal{I}';H)\)-valued functions.

Recall that \(\mathfrak m\) is supported on \(\varepsilon \mathbf{Q}^d\) and \(\varepsilon \leq (4r N^{2r})^{-1}\) holds by 7 . Let \(\phi \colon \mathbb{R}^d \to [0,1]\) be a fixed smooth function supported on \(\frac{5}{3}\mathbf{Q}^d\) and identically equal to \(1\) on \(\frac{4}{3}\mathbf{Q}^d\). Let \(\psi\) be a fixed nonnegative smooth function supported on \(\frac{1}{3} \mathbf{Q}^d\), normalized by \(\int_{\mathbb{R}^d} \psi(x)\, \mathrm{d}x = 1\).

We define \(\zeta \mathrel{\vcenter{:}}= \phi * \psi\). Note that \(\zeta \colon \mathbb{R}^d \to [0,1]\) is a smooth function supported on \(2 \mathbf{Q}^d\) and identically equal to \(1\) on \(\mathbf{Q}^d\). We also define \(\Phi_\varepsilon(x) \mathrel{\vcenter{:}}= \Phi(x/\varepsilon)\) for \(\Phi \in \{\phi, \psi, \zeta\}\).

Fix \(q \in {\mathcal{S}}_{L}\) and \(u' \in \mathcal{I}'\). We use the function \(\zeta_\varepsilon\) to obtain the following factorization \[\llbracket G \mid \mathfrak m\rrbracket_{\mathcal{R}(b_q) + u'} = \llbracket 1 \mid \mathfrak m\rrbracket_{\mathcal{Q}(b_q) + u'}\cdot \llbracket G \mid \zeta_{\varepsilon}\rrbracket_{\mathcal{R}(b_q) + u'}.\] The multiplier \(\llbracket 1 \mid \mathfrak m\rrbracket_{\mathcal{Q}(b_q) + u'}\) corresponds, up to modulation determined by \(u'\), to the operator \(T_{\mathbb{Z}^d}\big[\llbracket 1 \mid \mathfrak m\rrbracket_{\mathcal{Q}(b_q)}\big]\) which is bounded on \(\ell^{2r}(\mathbb{Z}^{d};H)\). In fact, by \(b_q \leq N\) and \(\varepsilon \leq (4r N^{2r})^{-1}\), we can use Theorem 3 to bound its norm by \({\boldsymbol{C}}_{\rm MSW}(d)\). By modulation symmetry of \(\mathcal{F}_{\mathbb{Z}^d}\) and Theorem 4 applied with \(f^{\zeta_\varepsilon}_{\mathcal{R}(b_q)+u'}\) in place of \(F_n\), we obtain 27 .

3.8 Square function estimates for the nice bump multiplier↩︎

Finally, for a fixed choice of admissible divisors \(b_q \in \mathcal{B}_q\) and for each \(s\in[1, \infty]\), we prove the following inequality \[\begin{align} \label{IWeq:13} \bigg [ \sum_{x \in \mathbb{Z}^d} \sum_{q \in \mathcal{S}_L} \Big( \sum_{u' \in \mathcal{I}' } \big \| f^{\zeta_\varepsilon}_{\mathcal{R}(b_q)+u'}(x) \big\|_{H}^{2} \Big)^s \bigg ]^{1/2s} \lesssim_d N^{C / \log \log N} \mathbf{C}_{G}(\mathcal{R}^d_{\leq N}) \|f\|_{\ell^{2s}(\mathbb{Z}^{d};H)}, \end{align}\tag{28}\] which we view as a linear operator norm bound from \(\ell^{2s}(\mathbb{Z}^{d};H)\) to \(\ell^{2s}(\mathbb{Z}^{d} \times \mathcal{S}_{L};\ell^{2}( \mathcal{I}' ;H))\). By interpolation, it suffices to prove 28 for \(s \in \{1, \infty\}\).

3.8.0.1 Case \(s=1\)

Each \(\llbracket G \mid \zeta_{\varepsilon}\rrbracket_{u + u'}\) is supported on \(u + u' + 2\varepsilon {\boldsymbol{Q}}^d\). We claim that these sets are pairwise disjoint. Indeed, for distinct \(q, q'\in {\mathcal{S}}_L\), the corresponding admissible divisors \(b_{q}, b_{q'} \in [N]\) are also distinct, since \(q=Q_{b_q} \neq Q_{b_{q'}}=q'\). Hence, the rational numbers \(u + u'\) appearing in 28 do not repeat. By 7 and \(r \geq 2\), we obtain \(\varepsilon \leq (4r N^{2r})^{-1} \le (8 N^{4})^{-1}\) and the claim follows, since the denominators of \(u + u'\) do not exceed \(N^2\). By Plancherel’s theorem 28 holds with \(\mathbf{C}_{G}(\mathcal{R}^d_{\leq N})\) in place of \(N^{C / \log \log N} \mathbf{C}_{G}(\mathcal{R}^d_{\leq N})\).

3.8.0.2 Case \(s=\infty\)

For every \(b_q \in \mathcal{B}_q\), we have the following inclusion–exclusion formula \[\sum_{u\in \mathcal{R}(b_q)} F(u) = \sum_{b \in \mathbb{Z}_+ : \, b|b_q} \mu(b_q / b) \sum_{u \in \mathcal{Q}(b)} F(u),\] where \(\mu \colon \mathbb{Z}_+ \to \{-1,0,1\}\) is the Möbius function. Now 28 for \(s=\infty\) follows from this decomposition with \(F(u)=f^{\zeta_\varepsilon}_{u+u'}\) for \(u\in \mathcal{R}(b_q)\) and Lemma 5, if only we can show \[\begin{align} \Big( \sum_{u' \in \mathcal{I}' } \big\| f^{\zeta_\varepsilon}_{\mathcal{Q}(b)+u'}(x) \big\|_{H}^2 \Big)^{1/2} \lesssim_d \mathbf{C}_{G}(\mathcal{R}^d_{\leq N}) \|f\|_{\ell^{\infty}(\mathbb{Z}^{d};H)} \end{align}\] with the implicit constant controlled uniformly in \(x \in \mathbb{Z}^d\) and \(b \in \bigcup_{q \in \mathcal{S}_L} \mathcal{B}_q\). By translation invariance, we can consider \(x=0\). By duality, it suffices to prove \[\begin{align} \label{5-squarefunction:3} \Big\| \sum_{u' \in \mathcal{I}' } h_{u'} \, \mathcal{F}_{\mathbb{Z}^d}^{-1} \big( \llbracket G \mid \zeta_{\varepsilon}\rrbracket_{\mathcal{Q}(b) + u'} \big) \Big\|_{\ell^1(\mathbb{Z}^{d};H)} \lesssim_d \mathbf{C}_{G}(\mathcal{R}^d_{\leq N}) \Big( \sum_{u' \in \mathcal{I}'} \|h_{u'}\|_H^2 \Big)^{1/2} \end{align}\tag{29}\] for any sequence \((h_{u'} : u'\in \mathcal{I}')\subseteq H\).

4 Proof of Theorem 2: the lower bound↩︎

In this section, we present a detailed proof of Theorem 2, illustrating that the dependence on \(N\) in Theorem 1 cannot be eliminated. This contrasts with Tao’s result [18] that establishes the Ionescu–Wainger multiplier theorem for the set of Ionescu–Wainger fractions with uniform bounds with respect to the size of the underlying family of fractions.

Our goal is to prove that, for \(p \in (1,\infty) \setminus \{2\}\) and infinitely many \(N \in \mathbb{Z}_+\), we have \[\label{eq:N-c95p} \big \| T_{\mathbb{Z}} \big[\llbracket 1 \mid {\mathfrak m}_{\varepsilon_N}\rrbracket_{\mathcal{R}_{\leq N}}\big] \big \|_{\ell^{p}(\mathbb{Z}) \to \ell^{p}(\mathbb{Z})} \gtrsim_{p} N^{(\log N)^{-c_p}}\tag{35}\] with some \(0 < \varepsilon_N \leq \vartheta_N\).

5 Seminorm variant of the Ionescu–Wainger theorem↩︎

In this section, we extend the Ionescu–Wainger multiplier theorem to the seminorm setting. The main objects will be the so-called Rademacher–Menshov seminorms, whose properties will be axiomatized below.

5.1 Seminorm variant of the Ionescu–Wainger multiplier theorem↩︎

The following result, which is the main result of this section, is a refinement of [19].

5.2 Examples of Rademacher–Menshov seminorms↩︎

Fix a Hilbert space \(H\) and parameters \(\rho \in [1,\infty]\) and \(\lambda \in \mathbb{R}_+\). For \(J \in \mathbb{Z}_+\) and \(\mathbb{I} \subseteq {\mathbb{N}}\), let \(\mathfrak S_J^{\mathbb{I}}\) be the space of all increasing sequences \(({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}}\) of numbers \({\mathfrak o}_j \in {\mathbb{I}}\). Let \((h_n : n \in {\mathbb{N}})\) be a sequence of elements of \(H\).

Given \(({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}} \in \mathfrak S_J^{\mathbb{I}}\), the \(\rho\)-oscillation seminorm of \((h_n : n\in \mathbb{I})\) is defined by \[\begin{align} O^{\rho}_{H; ({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}}}(h_n : n\in \mathbb{I}) \mathrel{\vcenter{:}}= \begin{cases} \Big(\displaystyle\sum_{j \in {\mathbb{N}}_{<J}} \displaystyle\sup_{n\in[{\mathfrak o}_j, {\mathfrak o}_{j+1}) \, \cap \, \mathbb{I}} \|h_n-h_{{\mathfrak o}_{j}} \|_{H}^{\rho} \Big)^{1/\rho}, & \rho \in \mathbb{R}_+,\\[1.5ex] \displaystyle \sup_{j \in {\mathbb{N}}_{<J}} \, \sup_{n\in[{\mathfrak o}_j, {\mathfrak o}_{j+1}) \, \cap \, \mathbb{I}} \|h_n-h_{{\mathfrak o}_{j}} \|_{H}, & \rho = \infty. \end{cases} \end{align}\] Similarly, the \(\rho\)-variation seminorm of \((h_n : n\in \mathbb{I})\) is defined by \[\begin{align} V_H^{\rho}(h_n : n\in\mathbb{I}) \mathrel{\vcenter{:}}= \begin{cases} \displaystyle\sup_{J\in\mathbb{N}} \, \sup_{({\mathfrak n}_j)_{j \in {\mathbb{N}}_{\leq J}} \in \mathfrak S_J^{\mathbb{I}}} \Big(\displaystyle \sum_{j \in {\mathbb{N}}_{< J}} \|h_{{\mathfrak n}_{j+1}}-h_{{\mathfrak n}_{j}}\|_{H}^{\rho} \Big)^{1/\rho}, & \rho \in \mathbb{R}_+, \\[1.5ex] \displaystyle \sup_{({\mathfrak n}_j)_{j \in {\mathbb{N}}_{\leq 1}} \in \mathfrak S_1^{\mathbb{I}}} \|h_{{\mathfrak n}_{1}}-h_{{\mathfrak n}_{0}}\|_{H}, & \rho = \infty. \end{cases} \end{align}\] Finally, the \(\lambda\)-jump counting function of \((h_n : n\in \mathbb{I})\) is defined by \[\begin{align} N_{H; \lambda} (h_n : n\in \mathbb{I}) \mathrel{\vcenter{:}}= \sup \big\{J\in \mathbb{Z}_+ : \min_{j \in {\mathbb{N}}_{< J}} \|h_{{\mathfrak n}_{j+1}}-h_{{\mathfrak n}_{j}}\|_{H} \geq \lambda \text{ for some } ({\mathfrak n}_j)_{j \in {\mathbb{N}}_{\leq J}} \in \mathfrak S_J^{\mathbb{I}} \big\}. \end{align}\] Here we use the convention that the supremum over the empty set equals \(0\). If \(H={\mathbb{C}}\), then we abbreviate \(O^{\rho}_{H; ({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}}}, V_{H}^{\rho}\), and \(N_{H; \lambda}\) to \(O^{\rho}_{({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}}}, V^{\rho}\), and \(N_{\lambda}\), respectively.

The above objects give rise to quasiseminorms on \(L^p\) spaces. Let \((X, \mathcal{B}, \mathfrak m)\) be a \(\sigma\)-finite measure space and fix \(p \in \mathbb{R}_+\). Let \((f_n : n \in {\mathbb{N}})\) be a sequence of functions \(f_n \in L^p(X;H)\).

The \(\rho\)-oscillation seminorm of \((f_n : n \in \mathbb{I})\) is defined by \[\begin{align} \label{eq:osc-Lp} \begin{aligned} \mathcal{O}_{H; X}^{\rho, p} (f_n : n \in \mathbb{I}) \mathrel{\vcenter{:}}= \sup_{J\in\mathbb{Z}_+} \, \sup_{({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}} \in \mathfrak S_J^{\mathbb{I}}} \big\| O^{\rho}_{H; ({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}}}(f_n : n\in \mathbb{I})\big\|_{L^{p}(X)}. \end{aligned} \end{align}\tag{44}\] Similarly, the \(\rho\)-variation seminorm of \((f_n : n \in \mathbb{I})\) is defined by \[\begin{align} \label{eq:var-Lp} \begin{aligned} \mathcal{V}_{H; X}^{\rho, p}(f_n : n\in \mathbb{I}) \mathrel{\vcenter{:}}= \big\| V_H^{\rho}(f_n : n\in \mathbb{I})\big\|_{L^{p}(X)}. \end{aligned} \end{align}\tag{45}\] Finally, the \(\rho\)-jump quasiseminorm of \((f_n : n \in \mathbb{I})\) is defined by \[\begin{align} \label{eq:jump-Lp} \begin{aligned} \mathcal{J}_{H; X}^{\rho, p}(f_n : n\in \mathbb{I}) \mathrel{\vcenter{:}}= \sup_{\lambda \in \mathbb{R}_+} \big\| \lambda \, N_{H; \lambda}(f_n : n\in \mathbb{I})^{1/\rho} \big\|_{L^{p}(X)}. \end{aligned} \end{align}\tag{46}\] As before, we use the convention that the supremum over the empty set equals \(0\). If \(H={\mathbb{C}}\), then we abbreviate \(\mathcal{O}_{H; X}^{\rho, p}, \mathcal{V}_{H; X}^{\rho, p}\), and \(\mathcal{J}_{H; X}^{\rho, p}\) to \(\mathcal{O}_{X}^{\rho, p}, \mathcal{V}_{X}^{\rho, p}\), and \(\mathcal{J}_{X}^{\rho, p}\), respectively.

We now formulate the following result, which has important practical significance.

6 Bourgain’s ergodic theorem↩︎

In the final section, we demonstrate how Theorem 1 and Theorem 5 can be used in practice. Specifically, we will give a short proof of Bourgain’s pointwise ergodic theorem with polynomial iterates.

Theorem 6 was proved by Bourgain in a series of groundbreaking papers [1]–[3], giving an affirmative answer to the so-called Bellow–Furstenberg problem, see [26] and [27]. In fact, mean convergence for 50 was established by Furstenberg [28] long before Bourgain’s papers [1]–[3]. Theorem 6 is a far-reaching generalization of Birkhoff’s pointwise ergodic theorem [29] and von Neumann’s mean ergodic theorem [30]. However, in contrast to Birkhoff’s result, almost everywhere convergence of 49 may fail for integrable functions \(f\in L^1(X)\) when \(\deg P \ge 2\), as shown by Buczolich and Mauldin in [31] for \(P(n)=n^2\) and by LaVictoire in [32] for \(P(n)=n^k\) with any integer \(k\ge 2\). More about this theorem and the ergodic context can be found in the second author’s survey [20] as well as in [33].

Here we prove a quantitative version of Theorem 6, which reads as follows.

Now a few comments about Theorem 6 and Theorem 7 are in order.

1. There are two ways to obtain pointwise convergence for \(A_{N;X, T}^{P}f\) from 49 .

   • Each of the inequalities from 51 –53 implies Theorem 6. We first obtain a \(\mu\)-almost everywhere limit along the set \(\mathbb{D}_{\tau}\). Since \(\tau \in (1,\infty)\) is arbitrary, using the positivity of \(A_{N;X, T}^{P}\) (i.e. \(A_{N;X, T}^{P}f\ge 0\) if \(f\ge 0\)) we obtain that the limit exists \(\mu\)-almost everywhere on \(X\) as \(N\to\infty\), see [33] for details.

   • The other approach is more flexible and applies to operators that are not necessarily positive. Namely, if we combine inequalities from 51 –53 for \(\tau=2\) with square function inequality 54 and invoke the upper bounds 47 and 48 , then the inequalities from 51 –53 hold with \(\mathbb{Z}_+\) in place of \(\mathbb{D}_{\tau}\) and so the implied constants in 51 –53 can be taken independent of \(\tau\). Now, if any of the inequalities from 51 –53 holds with \(\mathbb{Z}_+\) in place of \(\mathbb{D}_{\tau}\), we immediately see that the limit of \(A_{N;X, T}^{P}f\) exists \(\mu\)-almost everywhere on \(X\) as \(N\to\infty\).

2. Each of the inequalities from 51 –53 with \(\tau=2\) implies that for every \(p\in(1, \infty)\) there exists \(C_{p, P}\in\mathbb{R}_+\) such that, for every \(f\in L^p(X)\), we have \[\begin{align} \label{eq:11} \big\|\sup_{N\in\mathbb{Z}_+}|A_{N;X, T}^{P}f|\big\|_{L^p(X)} \le C_{p, P}\|f\|_{L^p(X)}. \end{align}\tag{55}\] Maximal inequality 55 , together with the pointwise convergence of \(A_{N;X, T}^{P} f\) and the dominated convergence theorem, implies convergence of \(A_{N;X, T}^{P} f\) in \(L^p(X)\) norm.

3. In view of the previous two items, Theorem 7 implies Theorem 6. The ranges of parameters in Theorem 7 are sharp. Inequality 51 was established in [14], inequality 52 was established in [15], whereas inequality 53 was established in [16]. Variants of the square function estimates in the spirit of 54 were studied in [14]. In fact, the authors in [14]–[16] studied multidimensional variants of 49 as well as its singular integral variants. Moreover, all inequalities from 51 –53 were established along \(\mathbb{Z}_+\) in place of \(\mathbb{D}_{\tau}\) and so the implied constants were shown to be independent of \(\tau\). Furthermore, they were also shown to be independent of the coefficients of the underlying polynomials.

4. Finally, we remark that inequality 52 implies 51 for all \(\rho \in (2, \infty)\). This was observed by Bourgain in [3], see also [23], [25]. From this point of view, inequality 52 can be thought of as an endpoint estimate for 51 . Interestingly, the bound \[O^{2}_{({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}}}(A_{N;X, T}^{P}f : N\in \mathbb{I})\lesssim V^{2}(A_{N;X, T}^{P}f : N\in \mathbb{I})\] holds pointwise for any \(({\mathfrak o}_j)_{j \in {\mathbb{N}}_{\leq J}}\) and \(J\in\mathbb{Z}_+\), and for any \(\mathbb{I}\subseteq \mathbb{Z}_+\), while we have \[\sup_{\|f\|_{L^p(X)}\le1}\mathcal{V}_{X}^{2, p}(A_{N;X, T}^{P}f: N\in \mathbb{D}_{2})=\infty.\] The authors in [16] showed that there are no obvious relations between oscillation inequality 53 and variation inequality 51 beyond the trivial pointwise bound. The latter suggests that oscillation inequality 53 should play a role of an endpoint for 51 similar to jump inequality 52 . Therefore, a natural question arises whether the bounds from 53 can be used to deduce estimates from 51 for all \(\rho \in (2, \infty)\).

We now focus on proving Theorem 7. In fact, we only prove inequalities 51 and 54 . Inequalities 52 and 53 can be derived in the same way as inequality 51 .

6.1 Preliminary reductions↩︎

First, we reduce the matter to the integer shift system \((\mathbb{Z}, \mathcal{B}(\mathbb{Z}), \mu_{\mathbb{Z}})\) endowed with the shift transformation \(S \colon \mathbb{Z}\to\mathbb{Z}\) defined by \(S(x) \mathrel{\vcenter{:}}= x-1\) for \(x\in\mathbb{Z}\). Here \(\mathcal{B}(\mathbb{Z})\) is the \(\sigma\)-algebra of all subsets of \(\mathbb{Z}\) and \(\mu_{\mathbb{Z}}\) is counting measure on \(\mathbb{Z}\). In this case, for every \(N \in \mathbb{Z}_+\), we shall abbreviate \(A_{N;\mathbb{Z}, S}^{P}f\) to \[\begin{align} \label{eq:13} A_{N;\mathbb{Z}}^{P}f(x) \mathrel{\vcenter{:}}= \frac{1}{N}\sum_{n\in [N]}f(x-P(n)), \qquad x\in \mathbb{Z}. \end{align}\tag{56}\] In view of the Calderón transference principle [4], see also [34], it suffices to prove Theorem 7 for the integer shift system. Then each of the inequalities 51 –54 for the integer shift system immediately implies its analog for an arbitrary measure-preserving system.

We also remark that the Calderón transference principle [4], [34] only transfers quantitative bounds (such as inequalities 51 –54 ) that imply pointwise almost everywhere convergence, but does not transfer pointwise almost everywhere convergence from the integer shift system to an arbitrary measure-preserving system. In fact, in the integer shift system, pointwise convergence is implied by norm convergence, since the \(\ell^\infty(\mathbb{Z})\) norm is dominated by the \(\ell^2(\mathbb{Z})\) norm. This is not necessarily true for other measure-preserving systems.

The advantage of working with the integer shift system is that the average \(A_{N;\mathbb{Z}}^{P}f\) can be expressed as the convolution of \(f\) with the kernel \[\begin{align} K_N^P\mathrel{\vcenter{:}}= \frac{1}{N}\sum_{n\in[N]}\mathbb{1}_{{\{P(n)\}}}. \end{align}\] The averages from 56 are discrete averaging operators of Radon type, and their integral counterparts are well understood through classical harmonic analysis methods, see [35] and also [36] or [24]. Using the Fourier transform, we have \[\begin{align} \mathcal{F}_{\mathbb{Z}}(A_{N; \mathbb{Z}}^{P} f)(\xi)=m_{N}(\xi)\mathcal{F}_{\mathbb{Z}}f(\xi), \qquad \xi\in\mathbb{T}, \end{align}\] where the multiplier \(m_N\) is a normalized exponential sum of the form \[\begin{align} m_{N}(\xi) \mathrel{\vcenter{:}}= \mathcal{F}_{\mathbb{Z}}K_N^P(\xi)= \frac{1}{N}\sum_{n\in[N]}e(-\xi\cdot P(n)). \end{align}\]

Here is the place where the classical circle method enters the game and can be used to understand the nature of the multiplier \(m_{N}\) by analyzing the major and minor arcs. This will require the concepts of canonical fractions and their corresponding major arcs.

Let \(\eta \colon \mathbb{R}\to[0, 1]\) be a smooth and even function satisfying \[\begin{align} \mathbb{1}_{{[-1/4, 1/4]}}\le \eta\le \mathbb{1}_{{(-1/2, 1/2)}}. \end{align}\] For any \(n, \xi\in\mathbb{R}\), we set \(\eta_{\le n}(\xi) \mathrel{\vcenter{:}}= \eta(2^{-n}\xi)\). For any \(l\in\mathbb{N}\), using the definition of canonical fractions 12 , we set \(\Sigma_{\leq l} \mathrel{\vcenter{:}}= \mathcal{R}_{\le 2^l}\) and \(\Sigma_l \mathrel{\vcenter{:}}= \Sigma_{\leq l} \setminus \Sigma_{\leq l-1}\). Note that \(|\Sigma_{\leq l}| \le 2^{2l}\). Similarly, for any \(l\in\mathbb{N}\) and \(m\in\mathbb{Z}\), we introduce the dyadic major arcs by \[\begin{align} {\mathcal{M}}_{\leq l, \leq m} \mathrel{\vcenter{:}}= \bigcup_{\theta\in \Sigma_{\leq l}}[\theta-2^m, \theta+2^m]. \end{align}\] We note that the sets \({\mathcal{M}}_{\le l, \leq m}\) are nondecreasing in both \(l\) and \(m\), and if \(m \le - 2l - 2\), then the arcs \([\theta - 2^m, \theta + 2^m]\) that comprise \({\mathcal{M}}_{\leq l, \leq m}\) are pairwise disjoint. We also define \[\begin{align} {\mathcal{M}}_{l, \leq m} \mathrel{\vcenter{:}}= {\mathcal{M}}_{\leq l, \leq m} \setminus {\mathcal{M}}_{\leq l-1, \leq m} \quad \text{and} \quad {\mathcal{M}}_{l,m} \mathrel{\vcenter{:}}= {\mathcal{M}}_{l,\leq m} \setminus {\mathcal{M}}_{l,\leq m-1}. \end{align}\]

Using 13 , we define the dyadic Ionescu–Wainger projections \(\Pi_{\leq l, \leq m} \colon \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})\) by \[\begin{align} \Pi_{\leq l, \leq m} f \mathrel{\vcenter{:}}= T_{\mathbb{Z}}\big[\llbracket 1 \mid \eta_{\le m}\rrbracket_{\Sigma_{\le l}}\big] f ,\qquad f \in \ell^2(\mathbb{Z}). \end{align}\] If \(m \le - 2l - 2\), then the following properties of the Ionescu–Wainger projections hold.

1. The operator \(\Pi_{\leq l, \leq m}\) is self-adjoint and real symmetric.

2. The function \(\mathcal{F}_{\mathbb{Z}} (\Pi_{\leq l, \leq m} f)\) is supported on the set \({\mathcal{M}}_{\leq l, \leq m}\).

3. If \(\mathcal{F}_{\mathbb{Z}} f\) vanishes on the set \({\mathcal{M}}_{\leq l, \leq m}\), then \(\Pi_{\leq l, \leq m} f \equiv 0\).

4. If \(\mathcal{F}_{\mathbb{Z}} f\) is supported on the set \({\mathcal{M}}_{\leq l, \leq m-2}\), then \(\Pi_{\leq l, \leq m} f = f\).

5. The operator \(\Pi_{\leq l, \leq m}\) is a contraction on \(\ell^2(\mathbb{Z})\).

6. Moreover, for any \(p\in(1, \infty)\) and \(\gamma\in(0, 1)\), there exists a constant \(C_p\in\mathbb{R}_+\) such that, for every \(l\in\mathbb{N}\) and \(m \in \mathbb{Z}\), if \(m \leq - 6 \max\{p, p/(p-1)\}(l+1)\), then \[\begin{align} \label{eq:116} \| \Pi_{\leq l, \leq m} f \|_{\ell^p(\mathbb{Z})} \lesssim_{p,\gamma} 2^{C_p \gamma l}\|f\|_{\ell^p(\mathbb{Z})}. \end{align}\tag{57}\]

Inequality 57 is a consequence of Theorem 1.

6.2 Minor arcs estimates↩︎

We fix \(p \in (1,\infty)\) and a polynomial \(P\in\mathbb{Z}[{\rm n}]\) with \(\deg P=d\ge 2\) . Then we choose \(p_0\in 2\mathbb{Z}_+\) such that \(p_0/(p_0-1)<p<p_0\). One may think that \(p_0\) is a sufficiently large even integer, which serves as the parameter \(p\) from inequality 57 and the parameter \(2r\) from Theorem 1 and Theorem 5. It ensures that inequality 57 as well as Theorem 1 and Theorem 5 can be applied. If necessary, \(p_0\in 2\mathbb{Z}_+\) may be further adjusted to the interpolation arguments that will be used throughout this section.

We fix a small parameter \[\begin{align} 0<\alpha<(10^6 dp_0)^{-1} \end{align}\] and a large integer \(C_0\in \mathbb{Z}_+\) depending on \(p_0\) and \(P\). For all \(N\in \mathbb{Z}_+\), we define the quantities \[\begin{align} l_{(N)} \mathrel{\vcenter{:}}= {\rm Log\,} N^\alpha \quad \text{and} \quad L_{(N)} \mathrel{\vcenter{:}}= {\rm Log\,} N-l_{(N)}, \end{align}\] using the log-scale notation \({\rm Log\,} N \mathrel{\vcenter{:}}= \lfloor \log_2 N\rfloor\). For each \(N \geq C_0\), we now decompose \(A_{N; \mathbb{Z}}^P f\) into minor-arc and major-arc pieces, respectively, as follows \[A_{N; \mathbb{Z}}^Pf=A_{N; \mathbb{Z}}^P (f -\Pi_{\leq l_{(N)}, \leq -d L_{(N)}} f) +A_{N; \mathbb{Z}}^P(\Pi_{\leq l_{(N)}, \leq -d L_{(N)}}f).\]

By inequality 57 , for any \(\gamma\in(0, 1)\), we have \[\begin{align} \label{eq:18} \big\|A_{N; \mathbb{Z}}^P(f -\Pi_{\leq l_{(N)}, \leq -d L_{(N)}} f) \big\|_{\ell^{p_0}(\mathbb{Z})} \lesssim_{\gamma} N^{\gamma} \|f\|_{\ell^{p_0}(\mathbb{Z})}. \end{align}\tag{58}\] This inequality can be improved by using Weyl’s inequality for the multiplier \(m_N\).

Inequality ?? is the classical Weyl sum estimate for normalized exponential sums, which can be deduced from its inverse form, see for instance [37].

Since \(\mathcal{F}_{\mathbb{Z}}(f -\Pi_{\leq l_{(N)}, \leq -d L_{(N)}} f)\) vanishes on the set \({\mathcal{M}}_{\leq l_{(N)}, \leq -d L_{(N)}}\), we can use Plancherel’s theorem and inequality ?? to conclude that there exists a constant \(c_2\in(0, 1)\) such that \[\begin{align} \label{eq:20} \big\|A_{N; \mathbb{Z}}^P(f -\Pi_{\leq l_{(N)}, \leq -d L_{(N)}} f) \big\|_{\ell^{2}(\mathbb{Z})} \lesssim N^{-c_2} \|f\|_{\ell^{2}(\mathbb{Z})}. \end{align}\tag{59}\]

By interpolating 58 with 59 and taking \(\gamma\in(0, 1)\) sufficiently small, we deduce that there exists a small absolute constant \(c_p\in(0, 1)\) such that, for all \(f\in \ell^p(\mathbb{Z})\), we have \[\begin{align} \label{eq:21} \big\|A_{N; \mathbb{Z}}^P(f -\Pi_{\leq l_{(N)}, \leq -d L_{(N)}} f) \big\|_{\ell^p(\mathbb{Z})} \lesssim N^{-c_p} \|f\|_{\ell^p(\mathbb{Z})}. \end{align}\tag{60}\] Let \(\mathbb{D}_{\tau,*} \mathrel{\vcenter{:}}= \{ N \in \mathbb{D}_{\tau} : N \geq C_0 \}\). Using 60 , we obtain \[\begin{align} \label{eq:22} \mathcal{V}_{\mathbb{Z}}^{\rho, p}\big(A_{N; \mathbb{Z}}^P(f -\Pi_{\leq l_{(N)}, \leq -d L_{(N)}} f) : N\in \mathbb{D}_{\tau,*}\big) \lesssim \sum_{N\in \mathbb{D}_{\tau,*}}N^{-c_p} \|f\|_{\ell^p(\mathbb{Z})}\lesssim_{\tau}\|f\|_{\ell^p(\mathbb{Z})}, \end{align}\tag{61}\] since \(\sum_{N\in \mathbb{D}_{\tau,*}}N^{-c_p}\lesssim_{\tau} 1\). Now it remains to understand the major-arc piece.

6.3 Major arcs estimates↩︎

By the splitting property and 61 , it remains to show \[\begin{align} \label{eq:23} \mathcal{V}_{\mathbb{Z}}^{\rho, p} \big(A_{N; \mathbb{Z}}^P(\Pi_{l, \leq -d L_{(N)}}f) :N\in\mathbb{D}_{\tau, l} \big) \lesssim 2^{-c_pl}\|f\|_{\ell^p(\mathbb{Z})} \end{align}\tag{62}\] for all \(l\in\mathbb{N}\) with \(\mathbb{D}_{\tau, l} \mathrel{\vcenter{:}}= \{N\in\mathbb{D}_{\tau}:N\ge C_0 \text{ and } N\ge 2^{l/\alpha}\}\). Indeed, since \[\begin{align} A_{N; \mathbb{Z}}^P(\Pi_{\leq l_{(N)}, \leq -d L_{(N)}}f)=\sum_{l \in {\mathbb{N}}_{\le l_{(N)}}} A_{N; \mathbb{Z}}^P(\Pi_{l, \leq -dL_{(N)}}f), \end{align}\] summing 62 over \(l\in\mathbb{N}\), and using the splitting property and 61 , we obtain 51 .

For each \(N\ge 1\), we define a continuous counterpart of \(m_N\) by \[\begin{align} \mathfrak{m}_N(\xi) \mathrel{\vcenter{:}}= \int_{0}^1e(-\xi\cdot P(Nt)) \,{\rm d}t, \qquad \xi\in\mathbb{R}. \end{align}\] For every \((a,q) \in\mathbb{Z} \times \mathbb{Z}_+\) such that \({\rm gcd}(a, q)=1\), we define the complete exponential sum \(G(a/q) \mathrel{\vcenter{:}}= m_q(a/q)\). A simple application of the mean value theorem (see [19]) shows that, for every \(\xi\in\mathbb{T}\) and \(\theta\in \Sigma_{l}\) such that \(|\xi-\theta|\le M^{-1}\) for some \(M \geq 1\), one has \[\begin{align} \label{eq:25} |m_N(\xi)-G(\theta)\mathfrak m_N(\xi-\theta)|\lesssim_P 2^{l} (M^{-1}N^{d-1}+N^{-1} ). \end{align}\tag{63}\]

By inequality 63 with \(M=2^{dL_{(N)}}\), there exists a constant \(c_p\in(0, 1)\) such that \[A_{N; \mathbb{Z}}^P(\Pi_{l, \leq -dL_{(N)}}f)=T_{\mathbb{Z}}\big[\llbracket G \mid \mathfrak m_N\eta_{\le -d L_{(N)}}\rrbracket_{\Sigma_{l}}\big]f+E_Nf,\] where \(\|E_Nf\|_{\ell^p(\mathbb{Z})} \lesssim N^{-c_p}\|f\|_{\ell^p(\mathbb{Z})}\). Arguing as in 61 , and using the last inequality for \(N\in\mathbb{D}_{\tau, l}\), we reduce the verification of inequality 62 to proving that \[\begin{align} \label{eq:26} \mathcal{V}_{\mathbb{Z}}^{\rho, p}\big(T_{\mathbb{Z}}\big[\llbracket G \mid \mathfrak m_N\eta_{\le -d L_{(N)}}\rrbracket_{\Sigma_{l}}\big]f:N\in\mathbb{D}_{\tau, l}\big) \lesssim 2^{-c_pl}\|f\|_{\ell^p(\mathbb{Z})}. \end{align}\tag{64}\] By Theorem 5, for every \(\gamma\in(0, 1)\), we have \[\begin{align} \label{eq:27} \mathcal{V}_{\mathbb{Z}}^{\rho, p_0}\big(T_{\mathbb{Z}}\big[\llbracket G \mid \mathfrak m_N\eta_{\le -d L_{(N)}}\rrbracket_{\Sigma_{l}}\big]f:N\in\mathbb{D}_{\tau, l}\big) \lesssim_{\gamma} 2^{\gamma l}\|f\|_{\ell^{p_0}(\mathbb{Z})}, \end{align}\tag{65}\] since, for every \(q\in(1, \infty)\), we have \[\begin{align} {\boldsymbol{A}}_{q} & \mathrel{\vcenter{:}}= \sup_{\omega \in \{-1,1\}^\mathbb{N}} \Big\|\sum_{n\in\mathbb{N}} \omega(n)\, T_{\mathbb{R}^d}[\mathfrak m_{\lfloor \tau^{n+1}\rfloor}\eta_{\le -d L_{(\lfloor \tau^{n+1}\rfloor)}}-\mathfrak m_{\lfloor \tau^{n}\rfloor}\eta_{\le -d L_{(\lfloor \tau^{n}\rfloor)}}]\Big\|_{L^{q}(\mathbb{R})\to L^{q}(\mathbb{R})}<\infty, \\ {\boldsymbol{B}}_{q} & \mathrel{\vcenter{:}}= \sup_{\|f\|_{L^q(\mathbb{R})}\le1}\mathcal{V}_{\mathbb{R}}^{\rho, q}\big(T_{\mathbb{R}}\big[\mathfrak m_{\lfloor \tau^{n}\rfloor}\eta_{\le -d L_{(\lfloor \tau^{n}\rfloor)}}\big]f:n\in{\mathbb{N}}\big)<\infty. \end{align}\] The bounds for \({\boldsymbol{A}}_{q}\) and \({\boldsymbol{B}}_{q}\) follow respectively from [36] and [25]. We also refer to [24] for a unified approach, which also allows us to handle jump and oscillation seminorms.

Inequality 65 can be improved by controlling the complete exponential sum \(G\) on \(\Sigma_{l}\). Indeed, define \(u\mathrel{\vcenter{:}}= 100(l+1)\) and note that \(N\ge \max\{2^{l/\alpha}, C_0\}\) for any \(N\in\mathbb{D}_{\tau, l}\). This implies \(N\ge 2^{10^5 dp_0l}C_0^{1/2}\ge 2^{100dp_0u}\), provided that \(C_0\ge 2^{10^6 dp_0}\). Thus, we may write \[\begin{align} \label{eq:28} T_{\mathbb{Z}}\big[\llbracket G \mid \mathfrak m_N\eta_{\le -d L_{(N)}}\rrbracket_{\Sigma_{l}}\big]f = T_{\mathbb{Z}}\big[\llbracket 1 \mid \mathfrak m_N\eta_{\le -d L_{(N)}}\rrbracket_{\Sigma_{\le l}}\big] \, T_{\mathbb{Z}}\big[\llbracket G \mid \eta_{\le -d up_0}\rrbracket_{\Sigma_{l}}\big]f. \end{align}\tag{66}\] By Plancherel’s theorem and the Weyl sum estimate for complete exponential sums, we immediately deduce that there exists a constant \(c\in(0, 1)\) such that \[\begin{align} \label{eq:29} \big\|T_{\mathbb{Z}}\big[\llbracket G \mid \eta_{\le -dp_0u }\rrbracket_{\Sigma_{l}}\big]f\big\|_{\ell^2(\mathbb{Z})}\lesssim 2^{-cl}\|f\|_{\ell^2(\mathbb{Z})}. \end{align}\tag{67}\] Combining 65 for \(p_0=2\) and \(\gamma=c/2\) with 67 and the identity 66 , we obtain \[\begin{align} \label{eq:30} \mathcal{V}_{\mathbb{Z}}^{\rho, 2}\big(T_{\mathbb{Z}}\big[\llbracket G \mid \mathfrak m_N\eta_{\le -d L_{(N)}}\rrbracket_{\Sigma_{l}}\big]f:N\in\mathbb{D}_{\tau, l}\big) \lesssim 2^{-c_2 l}\|f\|_{\ell^{2}(\mathbb{Z})} \end{align}\tag{68}\] with \(c_2 = c/2\). Interpolating 65 with 68 , and taking \(\gamma\in(0, 1)\) to be sufficiently small, we obtain inequality 64 for some \(c_p\in(0, 1)\). This completes the proof of inequality 51 .

6.4 Concluding remarks↩︎

As we remarked above, the proofs of inequalities 52 and 53 proceed with obvious changes, in much the same way as the proof of inequality 51 . We omit the details. We now briefly outline the proof of the square function estimate from 54 , which in the integer shift settings reads as follows \[\begin{align} \label{eq:31} \bigg\|\Big(\sum_{k\in{\mathbb{N}}}\sum_{m\in{\mathbb{N}}_{<2^l}}|\Delta_m A_{2^k+2^{k-l}m;\mathbb{Z}}^{P}f|^2\Big)^{1/2}\bigg\|_{\ell^p(\mathbb{Z})}\le C_{p, P}2^{-c_pl}\|f\|_{\ell^p(\mathbb{Z})}, \end{align}\tag{69}\] where \(\Delta_m A_{2^k+2^{k-l}m;\mathbb{Z}}^{P}\mathrel{\vcenter{:}}= A_{2^k+2^{k-l}(m+1);\mathbb{Z}}^{P}-A_{2^k+2^{k-l}m;\mathbb{Z}}^{P}\) for \(m\in{\mathbb{N}}\).

Since the left-hand side of 69 is defined by a square function, we can entirely rely on Theorem 1. However, the challenge is to gain decay in \(l \in \mathbb{N}\). For this purpose, the key observation explaining how the decay in \(l \in \mathbb{N}\) arises is that, for all \(p \in [1, \infty)\), we have \[\begin{align} \label{eq:16} \big\|A_{2^k+2^{k-l}(m+1);\mathbb{Z}}^{P}f-A_{2^k+2^{k-l}m;\mathbb{Z}}^{P}f\big\|_{\ell^p(\mathbb{Z})}\lesssim 2^{-l}\|f\|_{\ell^p(\mathbb{Z})}, \end{align}\tag{70}\] since \(2^k+2^{k-l}m\simeq 2^k\) for all \(m\in {\mathbb{N}}_{<2^l}\). To prove 69 , we will again analyze the minor and major arcs separately. On minor arcs, for every \(k\in{\mathbb{N}}\) and \(\gamma\in(0, 1)\), we have \[\begin{align} \label{eq:32} \bigg\|\Big(\sum_{m\in{\mathbb{N}}_{<2^l}}\big|\Delta_m A_{2^k+2^{k-l}m;\mathbb{Z}}^{P} (f-\Pi_{\leq l_{(2^k)}, \leq -d L_{(2^k)}}f ) \big|^2\Big)^{1/2}\bigg\|_{\ell^{p_0}(\mathbb{Z})}\lesssim_\gamma 2^{\gamma k}\|f\|_{\ell^{p_0}(\mathbb{Z})}, \end{align}\tag{71}\] which follows from 70 and inequality 57 . Combining 70 with 59 , we obtain \[\begin{align} \label{eq:33} \big\|\Delta_m A_{2^k+2^{k-l}m;\mathbb{Z}}^{P}(f-\Pi_{\leq l_{(2^k)}, \leq -d L_{(2^k)}}f ) \big\|_{\ell^{2}(\mathbb{Z})} \lesssim \min\{2^{-l}, 2^{-c_2k}\}\|f\|_{\ell^{2}(\mathbb{Z})}. \end{align}\tag{72}\] Using 72 and bounding the minimum by \(2^{-3l/4 - c_2k/4}\), we see that \[\begin{align} \label{eq:34} \bigg\|\Big(\sum_{m\in{\mathbb{N}}_{<2^l}}\big|\Delta_m A_{2^k+2^{k-l}m;\mathbb{Z}}^{P}(f-\Pi_{\leq l_{(2^k)}, \leq -d L_{(2^k)}}f )\big|^2\Big)^{1/2}\bigg\|_{\ell^{2}(\mathbb{Z})}\lesssim 2^{-l/4- c_2k/4}\|f\|_{\ell^{2}(\mathbb{Z})}. \end{align}\tag{73}\] Interpolating 73 with 71 and taking \(\gamma\in(0, 1)\) sufficiently small, and summing over \(k\in{\mathbb{N}}\), we obtain 69 with \(f-\Pi_{\leq l_{(2^k)}, \leq -d L_{(2^k)}}f\) in place of \(f\). Now the proof of inequality 69 is reduced to showing 69 with \(\Pi_{\leq l_{(2^k)}, \leq -d L_{(2^k)}}f\) in place of \(f\). For this purpose, we can mimic the argument from the major-arc estimates, as in the proof of inequality 51 . After major arcs estimates based on 63 with \(M=2^{dL_{(2^k)}}\), followed by an application of Theorem 1, we are reduced to showing that there exists a constant \(c_p\in(0, 1)\) such that \[\begin{align} \label{eq:35} \bigg\|\Big(\sum_{k\in{\mathbb{N}}}\sum_{m\in{\mathbb{N}}_{<2^l}}|T_{\mathbb{R}}\big[(\Delta_m \mathfrak m_{2^k+2^{k-l}m;\mathbb{Z}})\eta_{\le -d L_{(2^k)}}\big]f|^2\Big)^{1/2}\bigg\|_{L^p(\mathbb{R})}\lesssim 2^{-c_pl}\|f\|_{L^p(\mathbb{R})}. \end{align}\tag{74}\] Inequality 74 follows from square function considerations combined with the estimate \(\|T_{\mathbb{R}}[(\Delta_m \mathfrak m_{2^k+2^{k-l}m;\mathbb{Z}})]f\|_{L^p(\mathbb{R})}\lesssim 2^{-l}\|f\|_{L^p(\mathbb{R})}\), valid for all \(p\in[1, \infty)\), see [24] for details.

This completes the proof of inequality 69 and the proof of inequality 54 follows.

References↩︎

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1. Dariusz Kosz was partially supported by the National Science Centre of Poland grant SONATA BIS 2022/46/E/ST1/00036. Mariusz Mirek was partially supported by the NSF CAREER grant DMS-2236493. Wojciech Słomian was partially supported by AMS–Simons Travel Grant No. 00008780 and by the Foundation for Polish Science scholarship START 068.2023. For the purpose of Open Access the authors have applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission.↩︎