# Oscillation and jump inequalities for the polynomial ergodic averages along
multi-dimensional subsets of primes

Nathan Mehlhop and Wojciech Słomian  Department of Mathematics, Rutgers
University, Piscataway, NJ 08854-8019, USA<EMAIL_ADDRESS>BCAM -
Basque Center for Applied Mathematics, 48009 Bilbao, Spain & Faculty of Pure
and Applied Mathematics, Wrocław University of Science and Technology
Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland<EMAIL_ADDRESS>&
<EMAIL_ADDRESS>
###### Abstract.

We prove the uniform oscillation and jump inequalities for the polynomial
ergodic averages modeled over multi-dimensional subsets of primes. This is a
contribution to the Rosenblatt-Wierdl conjecture [32, Problem 4.12, p. 80]
with averages taken over primes. These inequalities provide endpoints for the
$r$-variational estimates obtained by Trojan [38].

###### Key words and phrases:

oscillation seminorm, jump inequality, ergodic average along primes

###### 2020 Mathematics Subject Classification:

37A30 (Primary), 37A46, 42B20

The authors were partially supported by the National Science Foundation (NSF)
grant DMS-2154712. The second author was supported by the Basque Government
through the BERC 2022–2025 program and by the Ministry of Science, Innovation
and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718.

##  1\. Introduction

The aim of this paper is to prove uniform oscillation inequalities and
$\lambda$-jump inequalities in the context of polynomial ergodic averages and
truncated singular operators of the Cotlar type modeled on multi-dimensional
subset of primes. We extend the known results of Trojan [38] for the
$r$-variation seminorm $V^{r}$ with $r>2$ to endpoint cases expressed in terms
of the uniform jump and oscillation inequalities. This provides a fuller
quantitative description of the pointwise convergence of the mentioned
averages.

###  1.1. Statement of results

Let $(X,\mathcal{B},\mu)$ be a $\sigma$-finite measure space endowed with a
family of invertible commuting and measure preserving transformations
$S_{1},\ldots,S_{d}:X\to X$. Let $\Omega$ be a bounded convex open subset of
$\mathbb{R}^{k}$ such that $B(0,c_{\Omega})\subseteq\Omega\subseteq B(0,1)$
for some $c_{\Omega}\in(0,1)$, where $B(0,u)$ is the open Euclidean ball in
$\mathbb{R}^{k}$ with radius $u>0$ centered at $0\in\mathbb{R}^{k}$. For any
$t>0$, we set

$\Omega_{t}:=\\{x\in\mathbb{R}^{k}:t^{-1}x\in\Omega\\}.$

We consider a polynomial mapping

(1.1)
$\mathcal{P}=(\mathcal{P}_{1},\dots,\mathcal{P}_{d})\colon\mathbb{Z}^{k}\to\mathbb{Z}^{d}$

where each $\mathcal{P}_{j}\colon\mathbb{Z}^{k}\to\mathbb{Z}$ is a polynomial
of $k$ variables with integer coefficients such that $\mathcal{P}_{j}(0)=0$.
Let $k^{\prime},k^{\prime\prime}\in\\{0,1,\ldots,k\\}$ with
$k=k^{\prime}+k^{\prime\prime}$. For $f\in L^{\infty}(X,\mu)$, we define the
associated ergodic averages by

(1.2)
$\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f(x):=\frac{1}{\vartheta_{\Omega}(t)}\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}f(S_{1}^{\mathcal{P}_{1}(n,p)}\cdots
S_{d}^{\mathcal{P}_{d}(n,p)}x)\mathds{1}_{\Omega_{t}}(n,p)\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)},\quad
x\in X,$

where $\pm\mathbb{P}$ denotes the set of positive and negative prime numbers
and

$\vartheta_{\Omega}(t):=\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}\mathds{1}_{\Omega_{t}}(n,p)\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)}$

is the Chebyshev function. We also consider the Cotlar type ergodic averages
given by

(1.3)
$\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f(x):=\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}f(S_{1}^{\mathcal{P}_{1}(n,p)}\cdots
S_{d}^{\mathcal{P}_{d}(n,p)}x)K(n,p)\mathds{1}_{\Omega_{t}}(n,p)\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)},\quad
x\in X,$

where $K\colon\mathbb{R}^{k}\setminus\\{0\\}\to\mathbb{C}$ is a
Calderón–Zygmund kernel satisfying the following conditions:

  1. (1)

The size condition: For every $x\in\mathbb{R}^{k}\setminus\\{0\\}$, we have

(1.4) $\lvert K(x)\rvert\lesssim\lvert x\rvert^{-k}.$

  2. (2)

The cancellation condition: For every $0<r<R<\infty$, we have

(1.5) $\int_{\Omega_{R}\setminus\Omega_{r}}K(y)\mathrm{d}y=0.$

  3. (3)

The Lipschitz continuity condition: For every
$x,y\in\mathbb{R}^{k}\setminus\\{0\\}$ with $2|y|\leq|x|$, we have

(1.6) $\lvert K(x)-K(x+y)\rvert\lesssim\lvert y\rvert\lvert x\rvert^{-(k+1)}.$

We recall the definitions of the oscillation seminorm and $\lambda$-jump
counting function. Let $\mathbb{I}\subseteq\mathbb{R}$. For an increasing
sequence $I=(I_{j}:j\in\mathbb{N})\subseteq\mathbb{I}$ and
$N\in\mathbb{N}\cup\\{\infty\\}$, the truncated oscillation seminorm of a
function $f\colon\mathbb{I}\to\mathbb{C}$ is defined by

(1.7) $\displaystyle
O_{I,N}^{2}(f(t):t\in\mathbb{I}):=\Big{(}\sum_{j=1}^{N}\sup_{\begin{subarray}{c}I_{j}\leq
t<I_{j+1}\\\ t\in\mathbb{I}\end{subarray}}\lvert
f(t)-f(I_{j})\rvert^{2}\Big{)}^{1/2}.$

For any $\lambda>0$ and $\mathbb{I}\subseteq\mathbb{R}$, the _$\lambda$ -jump
counting function_ of a function $f\colon\mathbb{I}\to\mathbb{C}$ is defined
by

(1.8) $\displaystyle
N_{\lambda}(f(t):t\in\mathbb{I}):=\sup\\{J\in\mathbb{N}\,|\,\exists_{\begin{subarray}{c}t_{0}<\dotsb<t_{J}\\\
t_{j}\in\mathbb{I}\end{subarray}}:\min_{0<j\leq J}\lvert
f(t_{j})-f(t_{j-1})\rvert\geq\lambda\\}.$

We can now state the main result of this paper.

######  Theorem 1.

Let $d,k\geq 1$ and let $\mathcal{P}$ be a polynomial mapping as in (1.1). Let
$k^{\prime},k^{\prime\prime}\in\\{0,1,\ldots,k\\}$ with
$k^{\prime}+k^{\prime\prime}=k$ and let
$\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ be either
$\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ or
$\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$. Then, for any
$p\in(1,\infty)$, there is a constant $C_{p,d,k,\deg\mathcal{P}}>0$ such that

(1.9) $\displaystyle\sup_{\lambda>0}\big{\lVert}\lambda
N_{\lambda}(\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f:t>0)^{1/2}\big{\rVert}_{L^{p}(X,\mu)}$
$\displaystyle\leq C_{p,d,k,\deg\mathcal{P}}\|f\|_{L^{p}(X,\mu)},$ (1.10)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f:t>0)\big{\rVert}_{L^{p}(X,\mu)}$
$\displaystyle\leq C_{p,d,k,\deg\mathcal{P}}\lVert f\rVert_{L^{p}(X,\mu)},$

for any $f\in L^{p}(X,\mu)$. Here, $\mathfrak{S}_{N}(\mathbb{R}_{+})$ is the
set of all strictly increasing sequences in $\mathbb{R}_{+}$ of length $N+1$
(see Section 2.2). The constant $C_{p,d,k,\deg\mathcal{P}}$ is independent of
the coefficients of the polynomial mapping $\mathcal{P}$.

In the proof of the above theorem, we use methods developed in [23, 26, 38]
and very recently in [20, 35]. We follow Bourgain’s approach [6] to use the
Calderón transference principle [7] which reduce the problem to the integer
shift system (see Section 2.3) and then exploit the Hardy–Littlewood circle
method to analyze the appropriate Fourier multipliers. The main tools used to
handle the estimates for the multiplier operators are: an appropriate
generalization of Weyl’s inequality (Proposition 6); the Ionescu–Wainger
multiplier theorem (see [12, 26] and [37]) combined with the
Rademacher–Menshov inequality (see [23]) and standard multiplier
approximations (Lemma 8); the Magyar–Stein–Wainger sampling principle [18] and
[24].

As a consequence of Theorem 1, we can state the following quantitative form of
the ergodic theorem concerning the averages
$\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ and
$\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$.

######  Corollary 2.

Let $(X,\mathcal{B},\mu)$ be a $\sigma$-finite measure space. Let $d,k\geq 1$
and let $\mathcal{P}$ be a polynomial mapping as in (1.1). Let
$k^{\prime},k^{\prime\prime}\in\\{0,1,\ldots,k\\}$ with
$k^{\prime}+k^{\prime\prime}=k$ and let
$\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ be either
$\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ or
$\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$. Let
$p\in(1,\infty)$ and $f\in L^{p}(X,\mu)$. Then we have:

  * (i)

(Mean ergodic theorem) the averages
$\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f$ converge in
$L^{p}(X,\mu)$ norm as $t\to\infty$;

  * (ii)

(Pointwise ergodic theorem) the averages
$\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f$ converge
pointwise $\mu$-almost everywhere on $X$ as $t\to\infty$;

  * (iii)

(Maximal ergodic theorem) the following maximal estimate holds:

(1.11)
$\displaystyle\big{\|}\sup_{t>0}|\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f|\big{\|}_{L^{p}(X,\mu)}\lesssim_{d,k,p,\deg\mathcal{P}}\|f\|_{L^{p}(X,\mu)};$

  * (iv)

(Oscillation ergodic theorem) the following uniform oscillation inequality
holds:

(1.12)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\|}O_{I,N}^{2}(\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f:t>0)\big{\|}_{L^{p}(X,\mu)}\lesssim_{d,k,p,\deg\mathcal{P}}\|f\|_{L^{p}(X,\mu)};$

  * (v)

(Variational ergodic theorem) for any $r\in(2,\infty)$, the following
$r$-variational inequality holds (see Section 2.2 for the definition of
$V^{r}$):

(1.13)
$\displaystyle\big{\|}V^{r}(\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f:t>0)\big{\|}_{L^{p}(X,\mu)}\lesssim_{d,k,p,r,\deg\mathcal{P}}\|f\|_{L^{p}(X,\mu)};$

  * (vi)

(Jump ergodic theorem) the following jump inequality holds:

(1.14) $\displaystyle\sup_{\lambda>0}\big{\|}\lambda
N_{\lambda}(\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f:t>0)^{1/2}\big{\|}_{L^{p}(X,\mu)}\lesssim_{d,k,p,\deg\mathcal{P}}\|f\|_{L^{p}(X,\mu)}.$

The implicit constants in (1.11), (1.12), (1.13), and (1.14) are independent
of the coefficients of the polynomial mapping $\mathcal{P}$.

A few comments are in order.

  1. (1)

Corollary 2 is the most general quantitative version of the one parameter
ergodic theorem for both averages
$\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ and
$\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ (cf. [21, Theorem
1.20]), which concludes the work of many authors over last decades – see
Section 1.2 for details.

  2. (2)

The mean ergodic theorem (i) easily follows from (ii) and (iii) by Lebesgue’s
dominated convergence theorem. Each inequality from (iv), (v), and (vi)
individually implies pointwise convergence (ii) and the maximal estimate
(iii). The jump inequality (vi) implies the variational ergodic theorem (v) in
the full range $r\in(2,\infty)$. Hence, the inequality (1.14) can be seen as
an $r=2$ endpoint for (1.13).

  3. (3)

Unfortunately, we do not know at this moment if the oscillation inequality
(1.12) is any kind of endpoint for the variational inequality (1.13). A recent
result from [20] shows that the oscillation estimates cannot be interpreted as
an endpoint in a way similar to how the jump inequalities are. See the
discussion in [20] and [21].

  4. (4)

The oscillation inequality (1.12) for the ergodic averages
$\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ can be seen as a
contribution to a problem posed by Rosenblatt and Wierdl [32, Problem 4.12, p.
80] in the early 1990’s about uniform oscillation inequalities for the
classical Birkhoff ergodic averages given by

(1.15) $\frac{1}{2N+1}\sum_{n=-N}^{N}f(S^{n}x).$

In 1998, Jones, Kaufman, Rosenblatt, and Wierdl [14] gave an affirmative
answer to this problem. The inequality (1.10) provides us with the uniform
oscillation inequality for the counterpart of (1.15) along the prime numbers
given by

$\frac{1}{2|\mathbb{P}_{N}|}\sum_{n=-N}^{N}{f(S^{n}x)\mathds{1}_{\mathbb{P}}(|n|)},$

where $\mathbb{P}_{N}=\mathbb{P}\cap[1,N]$. Moreover, the inequality (1.10) is
much more general than the originally posted problem since it concerns the
multi-dimensional averages along arbitrary polynomials with integer
coefficients.

  5. (5)

Parts (i), (ii), (iii), and (v) for the standard averages
$\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ with
$k^{\prime\prime}\geq 1$ in the presented generality were first obtained by
Trojan [38]. In the case with $k^{\prime\prime}=0$ (excluding the prime
numbers from the summation), the first proof of the variational inequality
(1.12) in the full range $r\in(2,\infty)$ was given by Mirek, Stein, and
Trojan [23].

  6. (6)

In the case of the Cotlar ergodic averages
$\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ with
$k^{\prime\prime}\geq 1$, the ergodic theorems (i), (ii), (iii) and (v) were
proven by Trojan [38] under the gradient condition

(1.16) $|x|^{k+1}|\nabla K(x)|\lesssim 1,$

but Trojan’s argument can be adapted with small changes to deal with
Calderón–Zygmund kernels which satisfy the more general condition (1.6). In
this case, the results for $k^{\prime\prime}\geq 1$ seem to be completely new.
For $\mathcal{H}_{t}^{\mathcal{P},k,0}$, the jump inequality was obtained by
Mirek, Stein, and Zorin-Kranich [26], and the oscillation ergodic theorem was
obtained by the second author [35].

  7. (7)

The oscillation inequality (1.12) and the jump inequality (1.14) are
completely new results for both types of averages when $k^{\prime\prime}\geq
1$ and follows by Theorem 1. When $k^{\prime\prime}=0$, the corresponding
results for the jump inequalities are known due to the work of Mirek, Stein,
and Zorin-Kranich [26]. The uniform oscillation inequality was proven by
Mirek, Słomian, and Szarek [20] in the case of the averages
$\mathcal{A}_{t}^{\mathcal{P},k,0}$ and by the second author [35] in the case
of $\mathcal{H}_{t}^{\mathcal{P},k,0}$.

###  1.2. Historical background

In 1931, Birkhoff [3] and von Neumann [30] proved that the averages

(1.17) $M_{N}f(x):=\frac{1}{N}\sum_{n=1}^{N}f(S^{n}x)$

converge pointwise $\mu$-almost everywhere on $X$ and in $L^{p}(X,\mu)$ norm
respectively for any $f\in L^{p}(X,\mu)$, $p\in[1,\infty)$, as $N\to\infty$.
In 1955, Cotlar [10] established the pointwise $\mu$-almost everywhere
convergence on $X$ as $N\to\infty$ of the ergodic Hilbert transform given by

$H_{N}f(x):=\sum_{1\leq|n|\leq N}\frac{f(S^{n}x)}{n}$

for any $f\in L^{p}(X,\mu)$. In 1968, Calderón [7] made an important
observation (now called the Calderón transference principle) that some results
in ergodic theory can be easily deduced from known results in harmonic
analysis. Namely, the convergence of the Birkhoff averages $M_{N}$ can be
deduced from the boundedness of the Hardy–Littlewood maximal function, and the
convergence of Cotlar’s averages $H_{N}$ follows from the boundedness of the
maximal function for the truncated discrete Hilbert transform. As we will see
ahead, this observation has had a huge impact in the study of convergence
problems in ergodic theory.

At the beginning of the 1980’s, Bellow [2] and independently Furstenberg [11]
posed the problem about pointwise convergence of the averages along squares
given by

$T_{N}f(x):=\frac{1}{N}\sum_{n=1}^{N}{f(S^{n^{2}}x)}.$

Despite its similarity to Birkhoff’s theorem, the problem of pointwise
convergence of the $T_{N}$ averages has a totally different nature from that
of its linear counterpart. In particular, the standard approach is
insufficient in this case.

We briefly sketch the classical approach of handling the problem of pointwise
convergence. It consists of two steps:

  * (a)

Establish $L^{p}$-boundedness for the corresponding maximal function.

  * (b)

Find a dense class of functions in $L^{p}(X,\mu)$ for which the pointwise
convergence holds.

In the case of Birkhoff’s averages $M_{N}$, the Calderón transference
principle allows one to deduce the estimate

$\|\sup_{N\in\mathbb{N}}|M_{N}f|\|_{L^{p}(X,\mu)}\lesssim_{p}\|f\|_{L^{p}(X,\mu)}$

for $p\in(1,\infty]$ from the estimate for the discrete Hardy–Littlewood
maximal function (and we have a weak-type estimate for $p=1$). In turn,
estimates for the discrete Hardy–Littlewood maximal function follow easily
from those for the continuous one. This establishes the first step (a). For
the second step, one can use the idea of Riesz decomposition [31] to analyze
the space ${\mathbb{I}}_{S}\oplus{\mathbb{T}}_{S}\subseteq L^{2}(X,\mu)$,
where

$\displaystyle\qquad\qquad{\mathbb{I}}_{S}:=\\{f\in L^{2}(X,\mu):f\circ
S=f\\}\qquad\text{ and }\qquad{\mathbb{T}}_{S}:=\\{h\circ S-h:h\in
L^{2}(X,\mu)\cap L^{\infty}(X,\mu)\\}.$

We see that $M_{N}f=f$ for $f\in{\mathbb{I}}_{S}$ and, for $g=h\circ
S-h\in{\mathbb{T}}_{S}$, we have

$M_{N}g(x)=\frac{1}{N}\big{(}h(S^{N+1}x)-h(Sx)\big{)}$

by telescoping. Consequently, we see that $M_{N}g\to 0$ as $N\to\infty$. This
establishes $\mu$-almost everywhere pointwise convergence of $M_{N}$ on
${\mathbb{I}}_{S}\oplus{\mathbb{T}}_{S}$, which is dense in $L^{2}(X,\mu)$.
Since $L^{2}(X,\mu)$ is dense in $L^{p}(X,\mu)$ for every $p\in[1,\infty)$,
this establishes (b).

In the case of the quadratic averages $T_{N}$, the matter is more complicated.
For the first step, by the Calderón transference principle, it is enough to
establish $\ell^{p}$ bounds for the maximal function given by

(1.18) $\sup_{N\in\mathbb{N}}\frac{1}{N}\sum_{n=1}^{N}f(x-n^{2}),\quad
f\in\ell^{p}(\mathbb{Z}).$

The $\ell^{p}$ estimate for the above maximal function does not follow
directly from the continuous counterpart and requires completely new methods.
However, a more serious problem arises in connection with the second step.
Namely, the idea of von Neumann fails in this case because the averages
$T_{N}g$ do not possess the telescoping property for $g\in\mathbb{T}_{S}$.

At the end of the 1980’s, Bourgain established the pointwise convergence of
the averages $T_{N}$ in a series of groundbreaking articles [4, 5, 6]. By
using the Hardy–Littlewood circle method from analytic number theory, he
established $\ell^{p}$-bounds for the maximal function (1.18), which
establishes step (a). He then bypassed the problem of finding the requisite
dense class of functions by using the oscillation seminorm (1.7). Bourgain [6]
proved that, for any $\lambda>1$ and any sequence of integers
$I=(I_{j}:{j\in\mathbb{N}})$ with $I_{j+1}>2I_{j}$ for all $j\in\mathbb{N}$,
we have

(1.19)
$\displaystyle\big{\lVert}O_{I,N}^{2}(T_{\lambda^{n}}f:n\in\mathbb{N})\big{\rVert}_{L^{2}(X,\mu)}\leq
C_{I,\lambda}(N)\lVert f\rVert_{L^{2}(X,\mu)},\qquad N\in\mathbb{N},$

for any $f\in L^{2}(X,\mu)$ with
$\lim_{N\to\infty}N^{-1/2}C_{I,\lambda}(N)=0$. This non-uniform inequality
(1.19) suffices to establish the pointwise convergence of the averaging
operators $T_{N}f$ for any $f\in L^{2}(X,\mu)$. In the same series of papers,
by similar methods, Bourgain established the pointwise convergence of the
averages along primes

$\frac{1}{|\mathbb{P}_{N}|}\sum_{n=1}^{N}f(S^{n}x)\mathds{1}_{\mathbb{P}}(n)$

for $f\in L^{p}(X,\mu)$ with $p>\frac{1}{2}(1+\sqrt{3})$. In the same year,
Wierdl [40] extended Bourgain’s result to $p\in(1,\infty)$.

In order to establish the inequality (1.19), Bourgain used the Hardy–Littewood
circle method and $r$-variation seminorms $V^{r}$. The $r$-variations were
introduced by Lépingle [17] in the context of families of bounded martingales.
In 1976, he proved that, for all $r\in(2,\infty)$, $p\in(1,\infty)$, and any
family of bounded martingales $(\mathfrak{f}_{n}\colon
X\to\mathbb{C}:n\in\mathbb{N})$, we have

$\displaystyle\lVert
V^{r}(\mathfrak{f}_{n}:n\in\mathbb{N})\rVert_{L^{p}(X)}\lesssim_{p,r}\sup_{n\in\mathbb{N}}\lVert\mathfrak{f}_{n}\rVert_{L^{p}(X)}$

with the implicit constant depending only on $p$ and $r$. The above inequality
is sharp in the sense that it fails for $r=2$, see [13] for a counterexample.

Bourgain observed that the $V^{r}$ seminorm can be used to obtain (1.19). This
is because, by Hölder’s inequality, we have

$O_{I,N}^{2}(T_{n}f:n\in\mathbb{N})\leq
N^{1/2-1/r}V^{r}(T_{n}f:n\in\mathbb{N})$

for $r\geq 2$. In order to prove the $r$-variational inequality for the
averages $T_{N}$, Bourgain used the $\lambda$-jump counting function. It can
easily be seen that

$\displaystyle\sup_{\lambda>0}\|\lambda
N_{\lambda}(T_{N}f:N\in\mathbb{N})^{1/2}\|_{L^{p}(X,\mu)}\leq\|V^{r}(T_{N}f:N\in\mathbb{N})\|_{L^{p}(X,\mu)}$

for every $r\geq 2$. The above inequality can be reversed in some sense [6].
Namely, for any $p\in(1,\infty)$ and any $r\in(2,\infty)$, we have

$\displaystyle\|V^{r}(T_{N}f:N\in\mathbb{N})\|_{L^{p,\infty}(X,\mu)}\lesssim_{p,r}\sup_{\lambda>0}\|\lambda
N_{\lambda}(T_{N}f:N\in\mathbb{N})^{1/2}\|_{L^{p,\infty}(X,\mu)}.$

For more details about oscillation, variation, and jump seminorms, we refer to
[20, 21] and [16].

The above arguments demonstrate that the problem of proving pointwise
convergence can be reduced to proving an appropriate $r$-variational estimate
or jump inequality. However, an intriguing question was the issue of
uniformity in the inequality (1.19). Shortly after the groundbreaking work of
Bourgain, Lacey [32, Theorem 4.23, p. 95] improved inequality (1.19) showing
that, for every $\lambda>1$, there is a constant $C_{\lambda}>0$ such that

(1.20)
$\displaystyle\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{L}_{\tau})}\big{\lVert}O_{I,N}^{2}(T_{\lambda^{n}}f:n\in\mathbb{N})\big{\rVert}_{L^{2}(X)}\leq
C_{\lambda}\lVert f\rVert_{L^{2}(X)},\quad f\in L^{2}(X,\mu),$

where $\mathbb{L}_{\tau}:=\\{\tau^{n}:n\in\mathbb{N}\\}$. This result
motivated the question about uniform estimates independent of $\lambda>1$ in
(1.20). In the case of Birkhoff’s averages, this question was explicitly
formulated in [32, Problem 4.12, p. 80].

In 1998, Jones, Kaufman, Rosenblatt, and Wierdl [14] established the uniform
oscillation inequality on $L^{p}(X,\mu)$ for the standard Birkhoff averages
$M_{N}$. Two years later, Campbell, Jones, Reinhold, and Wierdl [8]
established the uniform oscillation inequality for the ergodic Hilbert
transform. In 2003, Jones, Rosenblatt, and Wierdl [15] proved uniform
oscillation inequalities on $L^{p}(X,\mu)$ with $p\in(1,2]$ for the Birkhoff
averages over cubes. However, the case of polynomial averages, even one-
dimensional, was open until recent works [20, 35], and the case of averages
along primes was open until this paper.

In 2015, Mirek and Trojan [27], using the ideas of Bourgain and Wierdl,
established $\mu$-almost everywhere pointwise convergence of the Cotlar
averages along the primes,

$\sum_{p\in(\pm\mathbb{P}_{N})}\frac{f(S^{p})}{p}\log|p|.$

They proved that the corresponding maximal function is bounded on
$L^{p}(X,\mu)$ with $p>1$ and showed that the analogue of Bourgain’s non-
uniform oscillation inequality (1.19) holds for those averages.

In the same year, Zorin-Kranich [41] established the pointwise convergence of
the averages related to the polynomial mapping given by

$\tilde{\mathcal{P}}=(n,n^{2},n^{3},\ldots,n^{d})\colon\mathbb{Z}\to\mathbb{Z}^{d}.$

Namely, he proved that, for any $r>2$ and
$|\frac{1}{p}-\frac{1}{2}|<\frac{1}{2(d+1)}$, we have the following
$r$-variational estimate

$\|V^{r}(\mathcal{A}_{N}^{\tilde{\mathcal{P}},1,0}f:N\in\mathbb{N})\|_{L^{p}(X,\mu)}\lesssim_{p,r}\|f\|_{L^{p}(X,\mu)}.$

As a consequence, the averages $\mathcal{A}_{N}^{\tilde{\mathcal{P}},1,0}f$
converge $\mu$-almost everywhere for any $f\in L^{p}(X,\mu)$.

In 2016, Mirek and Trojan [28] established the pointwise convergence for the
averages (1.2) taken over cubes with $k^{\prime}=k$, that is

$A_{N,{\rm
cube}}^{\mathcal{P},k,0}f(x):=\frac{1}{N^{k}}\sum_{y\in[0,N]^{k}\cap\mathbb{Z}^{k}}f(S_{1}^{\mathcal{P}_{1}(y)}S_{2}^{\mathcal{P}_{1}(y)}\cdots
S_{d}^{\mathcal{P}_{d}(y)}x).$

There, Mirek and Trojan noted for the first time that the Rademacher–Menshov
inequality (2.5) may be used to establish $r$-variational estimates. For
$p\in(1,\infty)$ and $r>\max\\{p,p/(p-1)\\}$, they proved that

$\|V^{r}(A_{N,{\rm
cube}}^{\mathcal{P},k,0}f:N\in\mathbb{N})\|_{L^{p}(X,\mu)}\leq C_{p,d,k,{\rm
deg}\mathcal{P}}\|f\|_{L^{p}(X,\mu)}.$

Unfortunately, the methods introduced by Bourgain had limitations. These work
perfectly fine in the case of the $L^{2}$ estimates, but, in the case of an
$L^{p}$ estimates with $p\neq 2$, there arise difficulties which are hard to
overcome concerning the fractions around which major arcs are defined.
However, Ionescu and Wainger [12], in their groundbreaking 2005 work about
discrete singular Radon operators, introduced a set of fractions for which the
circle method can be applied towards $L^{p}$ estimates with $p\neq 2$.

In 2015, Mirek [19] built a discrete counterpart of the Littlewood–Paley
theory using the Ionescu–Wainger multiplier theorem and used it to reprove the
main result from [12]. In 2017, Mirek, Stein, and Trojan [22, 23] further
exploited these ideas together with the Rademacher–Menshov inequality from
[28] to obtain an $L^{p}$ estimate for the $r$-variation seminorm for both
$\mathcal{A}_{t}^{\mathcal{P},k,0}$ and $\mathcal{H}_{t}^{\mathcal{P},k,0}$
associated with convex sets in the full range of parameters. Namely, they
showed that

(1.21)
$\big{\|}V^{r}(\mathcal{M}_{t}^{\mathcal{P},k,0}f:t>0)\big{\|}_{L^{p}(X,\mu)}\lesssim_{d,k,p,r,\deg\mathcal{P}}\|f\|_{L^{p}(X,\mu)}$

for $p\in(1,\infty)$ and $r\in(2,\infty)$, where
$\mathcal{M}_{t}^{\mathcal{P},k,0}$ is either
$\mathcal{A}_{t}^{\mathcal{P},k,0}$ or $\mathcal{H}_{t}^{\mathcal{P},k,0}$.
There, the operators $\mathcal{H}_{t}^{\mathcal{P},k,0}$ are related to
Calderón–Zygmund kernels satisfying the gradient condition (1.16).

In 2019, Trojan [38] proved an $L^{p}$ estimate for the $r$-variation seminorm
for both $\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ and
$\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ with
$k^{\prime},k^{\prime\prime}\in\\{0,1,\ldots,k\\}$ such that
$k^{\prime}+k^{\prime\prime}=k$. Namely, he showed that

(1.22)
$\big{\|}V^{r}(\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f:t>0)\big{\|}_{L^{p}(X,\mu)}\lesssim_{d,k,p,r,\deg\mathcal{P}}\|f\|_{L^{p}(X,\mu)}$

for $p\in(1,\infty)$ and $r\in(2,\infty)$, where
$\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ is either
$\mathcal{A}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ or
$\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$. A straightforward
consequence of the inequality (1.22) is the $\mu$-almost everywhere
convergence of the averages
$\mathcal{M}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f$. Again, the
operators $\mathcal{H}_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}$ there
are related to Calderón–Zygmund kernels satisfying the gradient condition
(1.16).

In 2020, Mirek, Stein, and Zorin-Kranich [26] further refined the methods
developed in [22, 23] and proved a uniform $L^{p}$ estimate for the
$\lambda$-jump counting function. They proved that

(1.23) $\sup_{\lambda>0}\big{\lVert}\lambda
N_{\lambda}(\mathcal{M}_{t}^{\mathcal{P},k,0}f:t>0)^{1/2}\big{\rVert}_{L^{p}(X,\mu)}\leq
C_{p,d,k,{\rm deg}\mathcal{P}}\|f\|_{L^{p}(X,\mu)}$

for any $p\in(1,\infty)$ and any $f\in L^{p}(X,\mu)$, where
$\mathcal{M}_{t}^{\mathcal{P},k,0}f$ is either
$\mathcal{A}_{t}^{\mathcal{P},k,0}f$ or $\mathcal{H}_{t}^{\mathcal{P},k,0}f$.
There, the operators $\mathcal{H}_{t}^{\mathcal{P},k,0}$ are associated with
Calderón–Zygmund kernels satisfying the Hölder continuity condition
generalizing (1.6): For some $\sigma\in(0,1]$ and for every
$x,y\in\mathbb{R}^{k}\setminus\\{0\\}$ with $2|y|\leq|x|$, we have

(1.24) $\lvert K(x)-K(x+y)\rvert\lesssim\lvert y\rvert^{\sigma}\lvert
x\rvert^{-(k+\sigma)}.$

It is worth noting that the inequality (1.23) implies the $r$-variation
inequality (1.21).

In 2021, the second author in collaboration with Mirek and Szarek [20]
established the oscillation inequality

(1.25)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\mathcal{A}_{t}^{\mathcal{P},k,0}f:t>0)\big{\rVert}_{L^{p}(X,\mu)}\leq
C_{p,d,k,{\rm deg}\mathcal{P}}\lVert f\rVert_{L^{p}(X,\mu)},$

and, recently, the second author [35] proved the counterpart of (1.25) in the
case of the operators $\mathcal{H}_{t}^{\mathcal{P},k,0}$ related to
Calderón–Zygmund kernels satisfying (1.24).

##  2\. Notation and necessary tools

###  2.1. Basic notation

We denote $\mathbb{N}:=\\{1,2,\ldots\\}$,
$\mathbb{N}_{0}:=\\{0,1,2,\ldots\\}$, and $\mathbb{R}_{+}:=(0,\infty)$. For
$d\in\mathbb{N}$, the sets $\mathbb{Z}^{d}$, $\mathbb{R}^{d}$,
$\mathbb{C}^{d}$, and
$\mathbb{T}^{d}=(\mathbb{R}/\mathbb{Z})^{d}\equiv[-1/2,1/2)^{d}$ have the
standard meanings. For each $N\in\mathbb{N}$, we set

$\mathbb{N}_{N}:=\\{1,\ldots,N\\}.$

For any $x\in\mathbb{R}$, we set

$\lfloor x\rfloor:=\max\\{n\in\mathbb{Z}:n\leq x\\}.$

For $u\in\mathbb{N}$, we define the set

$\displaystyle 2^{u\mathbb{N}}:=\\{2^{un}\colon n\in\mathbb{N}\\}.$

For two non-negative numbers $A$ and $B$, we write $A\lesssim B$ to indicate
that $A\leq CB$ for some $C>0$ that may change from line to line, and we may
write $\lesssim_{\delta}$ if the implicit constant depends on $\delta$.

We denote the standard inner product on $\mathbb{R}^{d}$ by $x\cdot\xi$.
Moreover, for any $x\in\mathbb{R}^{d}$, we denote the $\ell^{2}$-norm and the
maximum norm respectively by

$\displaystyle\lvert x\rvert:=\lvert x\rvert_{2}:=\sqrt{x\cdot x}\qquad\text{
and }\qquad|x|_{\infty}:=\max_{1\leq k\leq d}|x_{k}|.$

For a multi-index $\gamma=(\gamma_{1},\dots,\gamma_{k})\in\mathbb{N}^{k}_{0}$,
we abuse the notation to write $|\gamma|:=\gamma_{1}+\cdots+\gamma_{k}$. No
confusion should arise since all multi-indices will be denoted by $\gamma$.

###  2.2. Seminorms

Let $\mathbb{I}\subseteq\mathbb{R}$ and $\lambda>0$. For
$N\in\mathbb{N}\cup\\{\infty\\}$, we denote by $\mathfrak{S}_{N}(\mathbb{I})$
the family of all strictly increasing sequences of length $N+1$ contained in
$\mathbb{I}$. We already defined the oscillation seminorm (1.7) and the
$\lambda$-jump counting function (1.8) in the introduction. For any
$r\in[1,\infty)$, the $r$-variation seminorm $V^{r}$ of a function
$f\colon\mathbb{I}\to\mathbb{C}$ is defined by

(2.1)
$V^{r}(f(t):t\in\mathbb{I}):=\sup_{\begin{subarray}{c}t_{0}<t_{1}<\cdots<t_{J}\\\
t_{j}\in\mathbb{I}\end{subarray}}\Big{(}\sum_{j=1}^{J}\lvert
f(t_{j})-f(t_{j-1})\rvert^{r}\Big{)}^{1/r}.$

The $r$-variational seminorm controls the oscillation seminorm and the
$\lambda$-jump counting function. Indeed, by Hölder’s inequality, we have

(2.2) $O_{I,N}^{2}(f(t):t\in\mathbb{I})\leq
N^{1/2-1/r}V^{r}(f(t):t\in\mathbb{I})$

for any $N\in\mathbb{N}$, $I\in\mathfrak{S}_{N}(\mathbb{I})$, and $r\geq 2$.
Moreover, for any $\lambda>0$, we have

(2.3) $\lambda N_{\lambda}(f(t):t\in\mathbb{I})^{1/r}\leq
V^{r}(f(t):t\in\mathbb{I}).$

We adopt notation to simultaneously handle the oscillation seminorm and the
$\lambda$-jump counting function for the sake of brevity and to emphasize the
required properties. Let $E$ be either of $\mathbb{R}^{d}$ or $\mathbb{Z}^{d}$
with the usual measures and let $(f_{t}:t\in\mathbb{I})\subset L^{p}(E)$. We
write

$\mathcal{S}_{E}^{p}(f_{t}:t\in\mathbb{I})$

to represent either of the following quantities:

$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{I})}\big{\|}O_{I,N}^{2}(f_{t}(x):t\in\mathbb{I})\big{\|}_{L^{p}(E)}\quad\text{or}\quad\sup_{\lambda>0}\big{\|}\lambda
N_{\lambda}(f_{t}(x):t\in\mathbb{I})^{1/2}\big{\|}_{L^{p}(E)}.$

######  Proposition 3.

Let $p\in(1,\infty)$ and $\mathbb{I}\subseteq\mathbb{R}$. The seminorm
$\mathcal{S}_{E}^{p}$ is subadditive up to a positive constant, that is,

$\mathcal{S}_{E}^{p}(f_{t}+g_{t}:t\in\mathbb{I})\lesssim\mathcal{S}_{E}^{p}(f_{t}:t\in\mathbb{I})+\mathcal{S}_{E}^{p}(g_{t}:t\in\mathbb{I}),$

where the implied constant is independent of $\mathbb{I}$ and the families
$(f_{t}:t\in\mathbb{I})$ and $(g_{t}:t\in\mathbb{I})$.

The critical point is that the jump quasi-seminorm admits an equivalent
subadditive seminorm, see [26, Corollary 2.11].

######  Remark 2.4 (Rademacher–Menshov inequality).

By inequalities (2.2) and (2.3), we deduce that the Rademacher–Menshov
inequality [25, Lemma 2.5, p. 534] holds for $\mathcal{S}_{E}^{p}$. Namely,
for any $k,m\in\mathbb{N}$ with $k<2^{m}$ and any sequence of functions
$(f_{n}:n\in\mathbb{N})\subset L^{p}(E)$, we have

(2.5) $\mathcal{S}_{E}^{p}(f_{n}:k\leq n\leq
2^{m})\leq\sqrt{2}\Big{\lVert}\sum_{i=1}^{s}\Big{(}\sum_{j}{|f_{u_{j+1}^{i}}-f_{u_{j}^{i}}|^{2}}\Big{)}^{1/2}\Big{\rVert}_{L^{p}(E)},$

where each $[u_{j}^{i},u_{j+1}^{i})$ is a dyadic interval contained in
$[k,2^{m}]$ of the form $[j2^{i},(j+1)2^{i})$ for some $0\leq i\leq m$ and
$0\leq j\leq 2^{m-i}-1$.

For more information about the $\lambda$-jump counting function and the
oscillation and $r$-variation seminorms, we refer to [6, 16, 21, 24, 34].

###  2.3. Reductions: Calderón transference and lifting

By the Calderón transference principle [7], we may restrict attention to the
model dynamical system of $\mathbb{Z}^{d}$ equipped with the counting measure
and the shift operators $S_{j}\colon\mathbb{Z}^{d}\to\mathbb{Z}^{d}$ given by
$S_{j}(x_{1},\ldots,x_{d}):=(x_{1},\ldots,x_{j}-1,\ldots,x_{d})$. We denote
the corresponding averaging operators by

$A_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f(x)=\frac{1}{\vartheta_{\Omega}(t)}\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}f\big{(}x-\mathcal{P}(n,p)\big{)}\mathds{1}_{{\Omega_{t}}}(n,p)\Big{(}\prod_{j=1}^{k^{\prime\prime}}\log|p_{j}|\Big{)}$

and

$H_{t}^{\mathcal{P},k^{\prime},k^{\prime\prime}}f(x)=\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}f\big{(}x-\mathcal{P}(n,p)\big{)}K(n,p)\mathds{1}_{{\Omega_{t}}}(n,p)\Big{(}\prod_{j=1}^{k^{\prime\prime}}\log\lvert
p_{j}\rvert\Big{)}.$

Moreover, by a standard lifting argument, it suffices to prove Theorem 1 for a
canonical case of the polynomial mapping $\mathcal{P}$. Let $\mathcal{P}$ be a
polynomial mapping as in (1.1). We define

${\rm deg}\,\mathcal{P}:=\max\\{{\rm deg}\,\mathcal{P}_{j}:1\leq j\leq d\\}$

and consider the set of multi-indices

$\Gamma:=\big{\\{}\gamma\in\mathbb{N}_{0}^{k}\setminus\\{0\\}:0<|\gamma|\leq{\rm
deg}\,\mathcal{P}\big{\\}}$

equipped with the lexicographic order. We define the _canonical polynomial
mapping_ by

(2.6) $\mathbb{R}^{k}\ni
x=(x_{1},\dots,x_{k})\mapsto\mathcal{Q}(x):=(x^{\gamma}\colon\gamma\in\Gamma)\in\mathbb{R}^{\Gamma},$

where $x^{\gamma}=x_{1}^{\gamma_{1}}x_{2}^{\gamma_{2}}\cdots
x_{k}^{\gamma_{k}}$. By invoking the lifting procedure described in [22, Lemma
2.2] (see also [36, Section 11]), the following implies Theorem 1.

######  Theorem 4.

Let $k\in\mathbb{N}$, let $\Gamma\subset\mathbb{N}^{k}\setminus\\{0\\}$ be a
nonempty finite set, and let
$k^{\prime},k^{\prime\prime}\in\\{0,1,\ldots,k\\}$ with
$k^{\prime}+k^{\prime\prime}=k$. Let $M_{t}^{k^{\prime},k^{\prime\prime}}$ be
either $A_{t}^{\mathcal{Q},k^{\prime},k^{\prime\prime}}$ or
$H_{t}^{\mathcal{Q},k^{\prime},k^{\prime\prime}}$. For any $p\in(1,\infty)$,
there is a constant $C_{p,k,|\Gamma|}>0$ such that

(2.7)
$\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}(M_{t}^{k^{\prime},k^{\prime\prime}}f:t>0)\leq
C_{p,k,|\Gamma|}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

###  2.4. Fourier transform and Ionescu–Wainger multiplier theorem

Let $\mathbb{G}=\mathbb{R}^{d}$ or $\mathbb{G}=\mathbb{Z}^{d}$ and let
$\mathbb{G}^{*}$ denote the dual group of $\mathbb{G}$. For every
$z\in\mathbb{C}$, we set $\bm{e}(z):=e^{2\pi{\bm{i}}z}$, where
${\bm{i}}^{2}=-1$. Let $\mathcal{F}_{\mathbb{G}}$ denote the Fourier transform
on $\mathbb{G}$ defined for any $f\in L^{1}(\mathbb{G})$ by

$\displaystyle\mathcal{F}_{\mathbb{G}}f(\xi):=\int_{\mathbb{G}}f(x)\bm{e}(x\cdot\xi){\rm
d}\mu(x),\quad\xi\in\mathbb{G}^{*},$

where $\mu$ is the usual Haar measure on $\mathbb{G}$. For any bounded
function $\mathfrak{m}\colon\mathbb{G}^{*}\to\mathbb{C}$, we define the
corresponding Fourier multiplier operator by

(2.8) $\displaystyle
T_{\mathbb{G}}[\mathfrak{m}]f(x):=\int_{\mathbb{G}^{*}}\bm{e}(-\xi\cdot
x)\mathfrak{m}(\xi)\mathcal{F}_{\mathbb{G}}f(\xi){\rm d}\xi,\quad
x\in\mathbb{G}.$

Here, we assume that $f\colon\mathbb{G}\to\mathbb{C}$ is a compactly supported
function on $\mathbb{G}$ (and smooth if $\mathbb{G}=\mathbb{R}^{d}$) or any
other function for which (2.8) makes sense.

An indispensable tool in the proof of Theorem 4 is the vector-valued
Ionescu–Wainger multiplier theorem from [26, Section 2] with an improvement by
Tao [37].

######  Theorem 5.

For every $\varrho>0$, there exists a family $(P_{\leq N})_{N\in\mathbb{N}}$
of subsets of $\mathbb{N}$ such that:

  1. (i)

$\mathbb{N}_{N}\subseteq P_{\leq
N}\subseteq\mathbb{N}_{\max\\{N,e^{N^{\varrho}}\\}}$.

  2. (ii)

If $N_{1}\leq N_{2}$, then $P_{\leq N_{1}}\subseteq P_{\leq N_{2}}$.

  3. (iii)

If $q\in P_{\leq N}$, then all factors of $q$ also lie in $P_{\leq N}$.

  4. (iv)

$\operatorname{lcm}(P_{N})\leq 3^{N}$.

Furthermore, for every $p\in(1,\infty)$, there exists
$0<C_{p,\varrho,|\Gamma|}<\infty$ such that, for every $N\in\mathbb{N}$, the
following holds:

Let $0<\varepsilon_{N}\leq e^{-N^{2\varrho}}$ and let
$\mathbf{Q}:=[-1/2,1/2)^{\Gamma}$ be a unit cube. Let
$\mathfrak{m}\colon\mathbb{R}^{\Gamma}\to L(H_{0},H_{1})$ be a measurable
function supported on $\varepsilon_{N}\mathbf{Q}$ taking values in
$L(H_{0},H_{1})$, the space of bounded linear operators between separable
Hilbert spaces $H_{0}$ and $H_{1}$. Let $0\leq\mathbf{A}_{p}\leq\infty$ denote
the smallest constant such that

$\big{\|}T_{\mathbb{R}^{\Gamma}}[\mathfrak{m}]f\big{\|}_{L^{p}(\mathbb{R}^{\Gamma};H_{1})}\leq\mathbf{A}_{p}\|f\|_{L^{p}(\mathbb{R}^{\Gamma};H_{0})}$

for every function $f\in L^{2}(\mathbb{R}^{\Gamma};H_{0})\cap
L^{p}(\mathbb{R}^{\Gamma};H_{0})$. Then, the multiplier

$\Delta_{N}(\xi):=\sum_{b\in\Sigma_{\leq N}}\mathfrak{m}(\xi-b),$

where $\Sigma_{\leq N}$ is defined by

$\Sigma_{\leq
N}:=\Big{\\{}\frac{a}{q}\in\mathbb{Q}^{\Gamma}\cap\mathbb{T}^{\Gamma}:q\in
P_{\leq N}\text{ and }{\rm gcd}(a,q)=1\Big{\\}},$

satisfies

(2.9)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[\Delta_{N}]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma};H_{1})}\leq
C_{p,\varrho,|\Gamma|}(\log
N)\mathbf{A}_{p}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma};H_{0})}$

for every $f\in\ell^{p}(\mathbb{Z}^{\Gamma};H_{0})$, (cf. [37, Theorem 1.4]
which removes the factor of $\log N$ in the inequality (2.9)).

##  3\. Preliminaries

###  3.1. General results

In this section, we present some general results concerning the behavior of
exponential sums. The following proposition is an enhancement of the variant
of Weyl’s inequality due to Trojan [38, Theorem 2] that allows us to estimate
exponential sums related to a possibly non-differentiable function $\phi$,
(cf. [26, Theorem A.1]).

######  Proposition 6 (Weyl’s inequality).

Let $\alpha>0$, $k\in\mathbb{N}$, and let
$\Gamma\subset\mathbb{N}^{k}\setminus\\{0\\}$ be a nonempty finite set. Let
$\Omega^{\prime}\subseteq\Omega\subseteq B(0,N)\subset\mathbb{R}^{k}$ be
convex sets and let $\phi\colon\Omega\cap\mathbb{Z}^{k}\to\mathbb{C}$. There
is $\beta_{\alpha}>0$ such that, for any $\beta>\beta_{\alpha}$, if there is a
multi-index $\gamma_{0}\in\Gamma$ with

$\displaystyle\Big{\lvert}\xi_{\gamma_{0}}-\frac{a}{q}\Big{\rvert}\leq\frac{1}{q^{2}}$

for some coprime integers $a$ and $q$ with $1\leq a\leq q$ and $(\log
N)^{\beta}\leq q\leq N^{|\gamma_{0}|}(\log N)^{-\beta}$, then

$\displaystyle\Big{\lvert}\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}\bm{e}(\xi\cdot\mathcal{Q}(n,p))\phi(n,p)\mathds{1}_{\Omega\setminus\Omega^{\prime}}(n,p)\Big{\rvert}$
$\displaystyle\lesssim
N^{k}\log(N)^{-\alpha}\lVert\phi\rVert_{L^{\infty}(\Omega\setminus\Omega^{\prime})}$
$\displaystyle\,\,+N^{k}\sup_{\begin{subarray}{c}\lvert x-y\rvert\leq N(\log
N)^{-\alpha}\\\
x,y\in\Omega\setminus\Omega^{\prime}\end{subarray}}\lvert\phi(x)-\phi(y)\rvert.$

The implicit constant is independent of the function $\phi$, the variable
$\xi$, the sets $\Omega,\Omega^{\prime}$, and the numbers $a$, $q$, and $N$.

###### Proof.

We define
$\tilde{\phi}(n,p,A):=\phi(n,p)\mathds{1}_{(\Omega\setminus\Omega^{\prime})\cap
A}(n,p)$. We partition the cube $[-N,N]^{k}$ into $J\lesssim\log(N)^{k\alpha}$
cubes $Q_{j}$ with disjoint interiors and side lengths $CN\log(N)^{-\alpha}$
for some constant $C>0$. Let $(m_{j},p_{j})$ be a fixed element of
$Q_{j}\cap\Omega\setminus\Omega^{\prime}$. Since
$\mathds{1}_{\Omega}(x)\mathds{1}_{\Omega\setminus\Omega^{\prime}}(x)=\mathds{1}_{\Omega\setminus\Omega^{\prime}}(x)$
for any $x\in\mathbb{R}^{k}$, we have

(3.1)
$\displaystyle\Big{\lvert}\sum_{(n,p)}\bm{e}(\xi\cdot\mathcal{Q}(n,p))\tilde{\phi}(n,p,\Omega)\mathds{1}_{\Omega}(n,p)\Big{\rvert}\lesssim\sum_{j=1}^{J}\Big{\lvert}\sum_{(n,p)}\bm{e}(\xi\cdot\mathcal{Q}(n,p))\tilde{\phi}(n,p,Q_{j})\mathds{1}_{\Omega\cap
Q_{j}}(n,p)\Big{\rvert},$

where all sums are taken over
$(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}$. Let
$(m_{j},p_{j})$ be a fixed element of
$Q_{j}\cap\Omega\setminus\Omega^{\prime}$. We estimate the right hand side of
(3.1) by

$\displaystyle\sum_{j=1}^{J}\Big{\lvert}\sum_{(n,p)}\bm{e}(\xi\cdot\mathcal{Q}(n,p))\tilde{\phi}(m_{j},p_{j},Q_{j})$
$\displaystyle\mathds{1}_{\Omega\cap Q_{j}}(n,p)\Big{\rvert}$ $\displaystyle+$
$\displaystyle\sum_{j=1}^{J}\Big{\lvert}\sum_{(n,p)}\bm{e}(\xi\cdot\mathcal{Q}(n,p))\big{(}\tilde{\phi}(m_{j},p_{j},Q_{j})-\tilde{\phi}(n,p,Q_{j})\big{)}\mathds{1}_{\Omega\cap
Q_{j}}(n,p)\Big{\rvert}.$

By Trojan’s variant of Weyl’s inequality [38, Theorem 2], the first term is
bounded by

(3.2) $JN^{k}(\log
N)^{-\alpha^{\prime}}\lVert\phi\rVert_{L^{\infty}(\Omega\setminus\Omega^{\prime})}\lesssim
N^{k}(\log
N)^{k\alpha-\alpha^{\prime}}\lVert\phi\rVert_{L^{\infty}(\Omega\setminus\Omega^{\prime})}$

for any $\alpha^{\prime}>0$. Since $\mathds{1}_{\Omega\cap
Q_{j}}(n,p)=\mathds{1}_{\Omega^{\prime}\cap
Q_{j}}(n,p)+\mathds{1}_{(\Omega\setminus\Omega^{\prime})\cap Q_{j}}(n,p)$, the
second term is bounded by

(3.3) $N^{k}(\log
N)^{k\alpha-\alpha^{\prime}}\lVert\phi\rVert_{L^{\infty}(\Omega\setminus\Omega^{\prime})}+N^{k}\sup_{\begin{subarray}{c}\lvert
x-y\rvert\leq N(\log N)^{-\alpha}\\\
x,y\in\Omega\setminus\Omega^{\prime}\end{subarray}}\lvert\phi(x)-\phi(y)\rvert.$

Choosing an appropriate $\alpha^{\prime}>0$ in (3.2) and (3.3) yields the
claim. ∎

The next result is a generalization of [38, Proposition 4.1] and [38,
Proposition 4.2] in the spirit of [26, Proposition 4.18]. For $q\in\mathbb{N}$
and $a\in\mathbb{N}_{q}^{\Gamma}$ with ${\rm gcd}(a,q)=1$, the _Gaussian sum_
related to the polynomial mapping $\mathcal{Q}$ is given by

(3.4)
$G(a/q):=\frac{1}{q^{k^{\prime}}}\frac{1}{\varphi(q)^{k^{\prime\prime}}}\sum_{x\in\mathbb{N}^{k^{\prime}}_{q}}\sum_{y\in
A_{q}^{k^{\prime\prime}}}\bm{e}((a/q)\cdot\mathcal{Q}(x,y)),$

where $A_{q}:=\\{a\in\mathbb{N}_{q}:{\rm gdc}(a,q)=1\\}$ and $\varphi$ is
Euler’s totient function. There is $\delta>0$ such that

(3.5) $\big{|}G(a/q)\big{|}\lesssim q^{-\delta},$

according to [38, Theorem 3].

######  Lemma 7.

Let $N\in\mathbb{N}$ and let $\Omega\subseteq B(0,N)\subset\mathbb{R}^{k}$ be
a convex set or a Boolean combination of finitely many convex sets. Let
$\mathcal{K}\colon\mathbb{R}^{k}\to\mathbb{C}$ be a continuous function
supported in $\Omega$. Then, for each $\beta>0$, there is a constant
$c=c_{\beta}>0$ such that, for any $q\in\mathbb{N}$ with $1\leq q\leq(\log
N)^{\beta}$, $a\in A_{q}$, and $\xi=a/q+\theta\in\mathbb{R}^{\Gamma}$, we have

$\displaystyle\bigg{|}\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}\bm{e}\big{(}\xi\cdot\mathcal{Q}(n,p)\big{)}\mathcal{K}(n,p)\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)}-G(a/q)\int_{\Omega}\bm{e}\big{(}(\xi-a/q)\cdot\mathcal{Q}(t)\big{)}\mathcal{K}(t){\rm
d}t\bigg{|}$
$\displaystyle\lesssim\big{[}N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\big{(}1+\sum_{\gamma\in\Gamma}|\theta_{\gamma}|N^{|\gamma|}\big{)}+N^{k}\sup_{\begin{subarray}{c}x,y\in\Omega\\\
|x-y|\leq
q\sqrt{k}\end{subarray}}|\mathcal{K}(x)-\mathcal{K}(y)|\big{]}N\exp\big{(}-c\sqrt{\log
N}\big{)}.$

The implied constant is independent of $N,a,q,\xi$ and the kernel
$\mathcal{K}$.

###### Proof.

The case when $k=k^{\prime}$ was proven in [26, Proposition 4.18], so we
assume that $k>k^{\prime}$. Observe that, for a prime number $p$, $p\mid q$ if
and only if $(p\bmod q,q)>1$. Hence, for each
$s\in\\{1,\ldots,k^{\prime\prime}\\}$, we have

(3.6)
$\displaystyle\bigg{|}\sum_{n\in\mathbb{N}_{0}^{k^{\prime}}}\sum_{\begin{subarray}{c}r^{\prime\prime}\in\mathbb{N}_{q}^{k^{\prime\prime}}\\\
(r_{s}^{\prime\prime},q)>1\end{subarray}}\sum_{\begin{subarray}{c}p\in\mathbb{P}^{k^{\prime\prime}}\\\
p\equiv r^{\prime\prime}\bmod
q\end{subarray}}\bm{e}(\xi\cdot\mathcal{Q}(n,p))\mathcal{K}(n,p)\bigg{(}\prod_{j=1}^{k^{\prime\prime}}\log
p_{j}\bigg{)}\bigg{|}$ $\displaystyle\lesssim
N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\sum_{p\mid q}\log p\lesssim
N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\log q\lesssim
N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\log\log(N).$

To simplify the notation, for $(x,y)\in\mathbb{R}^{k}\setminus\\{0\\}$, we set
$F(x,y):=\bm{e}(\theta\cdot\mathcal{Q}(x,y))\mathcal{K}(x,y).$ For
$(n,p)\in\mathbb{N}^{k^{\prime}}\times\mathbb{P}^{k^{\prime\prime}}$ with
$n\equiv r^{\prime}\bmod q$ and $p\equiv r^{\prime\prime}\bmod q$, we have

$\displaystyle\xi_{\gamma}n^{\gamma^{\prime}}p^{\gamma^{\prime\prime}}\equiv\frac{a_{\gamma}}{q}n^{\gamma^{\prime}}p^{\gamma^{\prime\prime}}+\theta_{\gamma}n^{\gamma^{\prime}}p^{\gamma^{\prime\prime}}\equiv\frac{a_{\gamma}}{q}(r^{\prime})^{\gamma^{\prime}}(r^{\prime\prime})^{\gamma^{\prime\prime}}+\theta_{\gamma}n^{\gamma^{\prime}}p^{\gamma^{\prime\prime}}\pmod{1}.$

Therefore, we have
$\bm{e}(\xi\cdot\mathcal{Q}(n,p))=\bm{e}((a/q)\cdot\mathcal{Q}(r^{\prime},r^{\prime\prime}))\bm{e}(\theta\cdot\mathcal{Q}(n,p)),$
so then

(3.7)
$\displaystyle\sum_{n\in\mathbb{N}_{0}^{k^{\prime}}}\sum_{p\in\mathbb{P}^{k^{\prime\prime}}}\bm{e}(\xi\cdot\mathcal{Q}(n,p))$
$\displaystyle\mathcal{K}(n,p)\bigg{(}\prod_{j=1}^{k^{\prime\prime}}\log
p_{j}\bigg{)}$
$\displaystyle=\sum_{r^{\prime}\in\mathbb{N}^{k^{\prime}}_{q}}\sum_{r^{\prime\prime}\in
A_{q}^{k^{\prime\prime}}}\bm{e}((a/q)\cdot\mathcal{Q}(r^{\prime},r^{\prime\prime}))\sum_{\begin{subarray}{c}n\in\mathbb{N}_{0}^{k^{\prime}}\\\
n\equiv r^{\prime}\bmod
q\end{subarray}}\sum_{\begin{subarray}{c}p\in\mathbb{P}^{k^{\prime\prime}}\\\
p\equiv r^{\prime\prime}\bmod
q\end{subarray}}F(n,p)\bigg{(}\prod_{j=1}^{k^{\prime\prime}}\log
p_{j}\bigg{)}$
$\displaystyle+\mathcal{O}\Big{(}N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\log\log
N\Big{)},$

where the error term is the cost for making the summation for
$r^{\prime\prime}$ over $A_{q}^{k^{\prime\prime}}$ instead of
$N_{q}^{k^{\prime\prime}}$. Fix $u\in\mathbb{N}^{k^{\prime}}$,
$\tilde{p}\in\mathbb{P}^{k^{\prime\prime}-1}$, and $r_{1}^{\prime\prime}\in
A_{q}$. Then
$\left\\{v\in\mathbb{N}:(u,v,\tilde{p})\in\Omega\right\\}=\left(V_{0}+1,\ldots,V_{1}\right)$
for some $0\leq V_{0}\leq V_{1}\leq N$. By partial summation, we obtain

(3.8)
$\displaystyle\sum_{\begin{subarray}{c}p_{1}\in\mathbb{P}_{V_{1}}\setminus\mathbb{P}_{V_{0}}\\\
p_{1}\equiv r_{1}^{\prime\prime}\bmod q\end{subarray}}F(u,p_{1},\tilde{p})\log
p_{1}=\sum_{\begin{subarray}{c}v_{1}=V_{0}+1\\\ v_{1}\equiv
r_{1}^{\prime\prime}\bmod
q\end{subarray}}^{V_{1}}F(u,v_{1},\tilde{p})\mathds{1}_{{\mathbb{P}}}(v_{1})\log
v_{1}$
$\displaystyle=\vartheta(V_{1};q,r^{\prime\prime}_{1})F(u,V_{1},\tilde{p})-\vartheta(V_{0};q,r^{\prime\prime}_{1})F(u,V_{0}+1,\tilde{p})-\sum_{v_{1}=V_{0}+1}^{V_{1}-1}\vartheta(v_{1};q,r_{1}^{\prime\prime})[F(u,v_{1}+1,\tilde{p})-F(u,v_{1},\tilde{p})],$

where, for $x\geq 1$, we have set

$\vartheta(x;q,r):=\sum_{\begin{subarray}{c}p\in\mathbb{P}_{x}\\\ p\equiv
r\bmod q\end{subarray}}\log p.$

Similarly, we have

(3.9)
$\sum_{v_{1}=V_{0}+1}^{V_{1}}F(u,v_{1},\tilde{p})=V_{1}F(u,V_{1},\tilde{p})-V_{0}F(u,V_{0}+1,\tilde{p})-\sum_{v_{1}=V_{0}+1}^{V_{1}-1}v_{1}[F(u,v_{1}+1,\tilde{p})-F(u,v_{1},\tilde{p})].$

Furthermore, in view of the Siegel–Walfisz theorem ([33, 39], see also [29,
Corollary 11.21]), there are $C,c^{\prime}>0$ such that for all $x\geq 1$,
$(r,q)=1$ and $1\leq q\leq(\log x)^{\beta^{\prime}}$,

(3.10) $\bigg{|}\vartheta(x;q,r)-\frac{x}{\varphi(q)}\bigg{|}\leq
Cx\exp\big{(}-c^{\prime}\sqrt{\log x}\big{)}.$

Hence, by (3.8), (3.9), and (3.10), we obtain

$\displaystyle\bigg{|}\sum_{\begin{subarray}{c}p_{1}\in\mathbb{P}_{V_{1}}\setminus\mathbb{P}_{V_{0}}\\\
p_{1}\equiv r_{1}^{\prime\prime}\bmod q\end{subarray}}F(u,p_{1},\tilde{p})\log
p_{1}-\frac{1}{\varphi(q)}\sum_{v_{1}=V_{0}+1}^{V_{1}}F(u,v_{1},\tilde{p})\bigg{|}$
$\displaystyle\qquad\lesssim\|\mathcal{K}\|_{L^{\infty}(\Omega)}\bigg{|}\vartheta(V_{1};q,r^{\prime\prime}_{1})-\frac{V_{1}}{\varphi(q)}\bigg{|}+\|\mathcal{K}\|_{L^{\infty}(\Omega)}\bigg{|}\vartheta(V_{0};q,r^{\prime\prime}_{1})-\frac{V_{0}}{\varphi(q)}\bigg{|}$
$\displaystyle\qquad\phantom{\lesssim}+\bigg{[}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\sum_{\gamma\in\Gamma}|\theta_{\gamma}|N^{|\gamma|-1}+\sup_{\begin{subarray}{c}x,y\in\Omega\\\
|x-y|\leq
1\end{subarray}}|\mathcal{K}(x)-\mathcal{K}(y)|\bigg{]}\sum_{v_{1}=V_{0}+1}^{V_{1}-1}\bigg{|}\vartheta(v_{1};q,r^{\prime\prime}_{1})-\frac{v_{1}}{\varphi(q)}\bigg{|}$
$\displaystyle\qquad\lesssim\big{[}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\big{(}1+\sum_{\gamma\in\Gamma}|\theta_{\gamma}|N^{|\gamma|}\big{)}+N\sup_{\begin{subarray}{c}x,y\in\Omega\\\
|x-y|\leq
1\end{subarray}}|\mathcal{K}(x)-\mathcal{K}(y)|\big{]}N\exp\big{(}-c^{\prime}\sqrt{\log
N}\big{)}.$

Similar arguments applied to the sums over $p_{2},\ldots,p_{k^{\prime\prime}}$
give

(3.11)
$\displaystyle\bigg{|}\sum_{\begin{subarray}{c}u\in\mathbb{N}_{0}^{k^{\prime}}\\\
u\equiv r^{\prime}\bmod q\end{subarray}}$
$\displaystyle\sum_{\begin{subarray}{c}p\in\mathbb{P}^{k^{\prime\prime}}\\\
p\equiv r^{\prime\prime}\bmod
q\end{subarray}}F(u,p)\bigg{(}\prod_{j=1}^{k^{\prime\prime}}\log
p_{j}\bigg{)}-\frac{1}{\varphi(q)^{k^{\prime\prime}}}\sum_{u\in\mathbb{N}_{0}^{k^{\prime}}}\sum_{v\in\mathbb{N}^{k^{\prime\prime}}}F(qu+r^{\prime},v)\bigg{|}$
$\displaystyle\lesssim\big{[}N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\big{(}1+\sum_{\gamma\in\Gamma}|\theta_{\gamma}|N^{|\gamma|}\big{)}+N^{k}\sup_{\begin{subarray}{c}x,y\in\Omega\\\
|x-y|\leq
1\end{subarray}}|\mathcal{K}(x)-\mathcal{K}(y)|\big{]}N\exp\big{(}-c^{\prime}\sqrt{\log
N}\big{)}.$

Let
$\Omega_{+}:=\Omega\cap[0,\infty)^{k^{\prime}}\times[1,\infty)^{k^{\prime\prime}}$.
We can estimate the sum by an integral by writing

(3.12)
$\displaystyle\bigg{|}q^{k^{\prime}}\sum_{u\in\mathbb{N}_{0}^{k^{\prime}}}\sum_{v\in\mathbb{N}^{k^{\prime\prime}}}F(qu+r^{\prime},v)-\iint_{\Omega_{+}}F(s,t)\textrm{d}s\textrm{d}t\bigg{|}$
$\displaystyle=\bigg{|}q^{k^{\prime}}\sum_{u\in\mathbb{N}_{0}^{k^{\prime}}}\sum_{v\in\mathbb{N}^{k^{\prime\prime}}}F(qu+r^{\prime},v)-\sum_{u\in\mathbb{N}_{0}^{k^{\prime}}}\sum_{v\in\mathbb{N}^{k^{\prime\prime}}}\int_{qu+[0,q)^{k^{\prime}}}\int_{v+[0,1)^{k^{\prime\prime}}}F(s,t)\textrm{d}s\textrm{d}t\bigg{|}$
$\displaystyle\leq\sum_{u\in\mathbb{N}_{0}^{k^{\prime}}}\sum_{v\in\mathbb{N}^{k^{\prime\prime}}}\int_{[0,q)^{k^{\prime}}}\int_{[0,1)^{k^{\prime\prime}}}|F(qu+r^{\prime},v)-F(qu+s,v+t)|\textrm{d}s\textrm{d}t.$

We use three estimates to control this:

$\displaystyle|\bm{e}(\theta\cdot\mathcal{Q}(qu+r^{\prime},v))-\bm{e}(\theta\cdot\mathcal{Q}(qu+s,v+t))|$
$\displaystyle\lesssim\sum_{\gamma\in\Gamma}q|\theta_{\gamma}|N^{|\gamma|-1},$
$\displaystyle|\mathcal{K}(qu+r^{\prime},v)-\mathcal{K}(qu+s,v+t)|$
$\displaystyle\lesssim\sup_{\begin{subarray}{c}x,y\in\Omega\\\ |x-y|\leq
q\sqrt{k}\end{subarray}}|\mathcal{K}(x)-\mathcal{K}(y)|,$
$\displaystyle\sum_{u\in\mathbb{N}_{0}^{k^{\prime}}}\sum_{v\in\mathbb{N}^{k^{\prime\prime}}}|\mathds{1}_{{\Omega}}(qu+r^{\prime},v)-\mathds{1}_{{\Omega}}(qu+s,v+t)|$
$\displaystyle\lesssim(N/q)^{k-1},$

where the last inequality is a consequence of [26, Proposition 4.16], which
gives that the number of lattice points in $\Omega$ at a distance $<q$ from
the boundary of $\Omega$ is $\mathcal{O}(qN^{k-1})$. We therefore get a bound
for (LABEL:eq:sum_int) of the form

$\displaystyle\mathcal{O}\bigg{(}qN^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\bigg{[}1+\sum_{\gamma\in\Gamma}|\theta_{\gamma}|N^{|\gamma|}\bigg{]}+N^{k}\sup_{\begin{subarray}{c}x,y\in\Omega\\\
|x-y|\leq q\sqrt{k}\end{subarray}}|\mathcal{K}(x)-\mathcal{K}(y)|\bigg{)}.$

Applying this in (3.11) and combining the error terms appropriately gives

$\displaystyle\bigg{|}\sum_{\begin{subarray}{c}u\in\mathbb{N}_{0}^{k^{\prime}}\\\
u\equiv r^{\prime}\bmod q\end{subarray}}$
$\displaystyle\sum_{\begin{subarray}{c}p\in\mathbb{P}^{k^{\prime\prime}}\\\
p\equiv r^{\prime\prime}\bmod
q\end{subarray}}F(u,p)\bigg{(}\prod_{j=1}^{k^{\prime\prime}}\log
p_{j}\bigg{)}-\frac{1}{q^{k^{\prime}}}\frac{1}{\varphi(q)^{k^{\prime\prime}}}\iint_{\Omega_{+}}F(s,t)\textrm{d}s\textrm{d}t\bigg{|}$
$\displaystyle\lesssim\big{[}N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\big{(}1+\sum_{\gamma\in\Gamma}|\theta_{\gamma}|N^{|\gamma|}\big{)}+N^{k}\sup_{\begin{subarray}{c}x,y\in\Omega\\\
|x-y|\leq
q\sqrt{k}\end{subarray}}|\mathcal{K}(x)-\mathcal{K}(y)|\big{]}N\exp\big{(}-c^{\prime}\sqrt{\log
N}\big{)}.$

Applying this in (3.7) by summing in $r^{\prime}$ and $r^{\prime\prime}$
together with (LABEL:eq:coprime) gives

(3.13)
$\displaystyle\bigg{|}\sum_{(n,p)\in\mathbb{N}_{0}^{k^{\prime}}\times\mathbb{P}^{k^{\prime\prime}}}\bm{e}\big{(}\xi\cdot\mathcal{Q}(n,p)\big{)}\mathcal{K}(n,p)\mathds{1}_{\Omega}(n,p)\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)}-G(a/q)\int_{\Omega_{+}}\bm{e}\big{(}(\xi-a/q)\cdot\mathcal{Q}(t)\big{)}\mathcal{K}(t){\rm
d}t\bigg{|}$
$\displaystyle\lesssim\big{[}N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\big{(}1+\sum_{\gamma\in\Gamma}|\theta_{\gamma}|N^{|\gamma|}\big{)}+N^{k}\sup_{\begin{subarray}{c}x,y\in\Omega\\\
|x-y|\leq
q\sqrt{k}\end{subarray}}|\mathcal{K}(x)-\mathcal{K}(y)|\big{]}N\exp\big{(}-c\sqrt{\log
N}\big{)}$

for any $c<c^{\prime}$. In simplifying to get the error term above, note that
$q^{k^{\prime}}\phi(q)^{k^{\prime\prime}}\leq q^{k}\leq(\log N)^{\beta k}$ and

$(\log N)^{\beta k}\exp(-c^{\prime}\sqrt{\log N})\lesssim\exp(-c\sqrt{\log
N}).$

Finally, we note that we can increase the range of integration at (3.13) to
the larger $\Omega\cap[0,\infty)^{k}$ by noting that

$G(a/q)\int_{\Omega\cap[0,\infty]^{k^{\prime}}\times[0,1)^{k^{\prime\prime}}}\bm{e}\big{(}(\xi-a/q)\cdot\mathcal{Q}(t)\big{)}\mathcal{K}(t){\rm
d}t$

is bounded by $N^{k^{\prime}}\|\mathcal{K}\|_{L^{\infty}(\Omega)}\leq
N^{k-1}\|\mathcal{K}\|_{L^{\infty}(\Omega)}$.

We can repeat the entire proof replacing $\mathbb{N}_{0}$ with
$-\mathbb{N}_{0}$ and/or $\mathbb{P}$ with $-\mathbb{P}$ in all the $2^{k}$
many possible combinations thereof in
$\mathbb{N}_{0}^{k^{\prime}}\times\mathbb{P}^{k^{\prime\prime}}$. Then,
collecting all of the error terms yields the claim. ∎

###  3.2. Multipliers for the averaging operators

For a function $f\colon\mathbb{Z}^{\Gamma}\rightarrow\mathbb{C}$ with finite
support, we have

$A_{t}^{\mathcal{Q},k^{\prime},k^{\prime\prime}}f(x)=T_{\mathbb{Z}^{\Gamma}}[\mathfrak{m}_{t}]f(x)\quad\text{and}\quad
H_{t}^{\mathcal{Q},k^{\prime},k^{\prime\prime}}f(x)=T_{\mathbb{Z}^{\Gamma}}[\mathfrak{n}_{t}]f(x)$

for the discrete Fourier multipliers

$\mathfrak{m}_{t}(\xi):=\frac{1}{\vartheta_{\Omega}(t)}\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}\bm{e}\big{(}\xi\cdot\mathcal{Q}(n,p)\big{)}\mathds{1}_{\Omega_{t}}(n,p)\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)},\quad\xi\in\mathbb{T}^{\Gamma},$

and

$\mathfrak{n}_{t}(\xi):=\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}\bm{e}(\xi\cdot\mathcal{Q}(n,p))K(n,p)\mathds{1}_{\Omega_{t}}(n,p)\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)},\quad\xi\in\mathbb{T}^{\Gamma}.$

Their continuous counterparts are given by

$\Phi_{t}(\xi):=\frac{1}{|\Omega_{t}|}\int_{\Omega_{t}}\bm{e}(\xi\cdot\mathcal{Q}(t)){\rm
d}t\quad\text{and}\quad\Psi_{t}(\xi):={\rm
p.v.}\int_{\Omega_{t}}\bm{e}(\xi\cdot\mathcal{Q}(t))K(t){\rm d}t$

respectively. To present a unified approach, we write
$M_{t}^{k^{\prime},k^{\prime\prime}}$, $\mathfrak{y}_{t}$, and $\Theta_{t}$ to
represent either $A_{t}^{\mathcal{Q},k^{\prime},k^{\prime\prime}}$,
$\mathfrak{m}_{t}$, and $\Phi_{t}$ or
$H_{t}^{\mathcal{Q},k^{\prime},k^{\prime\prime}}$, $\mathfrak{n}_{t}$, and
$\Psi_{t}$ respectively. We now present the key properties of our multiplier
operators that will be used in the proof of Theorem 4. Let $N_{n}:=\lfloor
2^{n^{\tau}}\rfloor$ for $n\in\mathbb{N}$ and some $\tau\in(0,1]$ adjusted
later.

  1. Property 1.

For each $\alpha>0$, there is $\beta_{\alpha}>0$ such that, for any
$\beta>\beta_{\alpha}$ and $n\in\mathbb{N}$, if there is a multi-index
$\gamma_{0}\in\Gamma$ with

$\bigg{|}\xi_{\gamma_{0}}-\frac{a}{q}\bigg{|}\leq\frac{1}{q^{2}}$

for some coprime integers $a$ and $q$ with $1\leq a\leq q$ and $(\log
N_{n})^{\beta}\leq q\leq N_{n}^{\lvert\gamma_{0}\rvert}(\log N_{n})^{-\beta}$,
then

$|(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})(\xi)|\lesssim C(\log
N_{n})^{-\alpha}.$

This follows from Proposition 6 with
$\phi(x)\equiv(\vartheta_{\Omega}(N_{n}))^{-1}$ for the
$\mathfrak{y}_{t}=\mathfrak{m}_{t}$ case and with $\phi(x)=K(x)$ for the
$\mathfrak{y}_{t}=\mathfrak{n}_{t}$ case, noting the size condition (1.4) and
the continuity condition (1.6).

  2. Property 2.

Let $A$ be the $|\Gamma|\times|\Gamma|$ diagonal matrix with

(3.14) $(Av)_{\gamma}=\lvert\gamma\rvert v_{\gamma}.$

For any $t>0$, we set
$t^{A}v:=\big{(}t^{\lvert\gamma\rvert}v_{\gamma}:\gamma\in\Gamma\big{)}.$ Then

$\big{|}\Theta_{N_{n}}(\xi)-\Theta_{N_{n-1}}(\xi)\big{|}\lesssim\min\big{\\{}|N_{n}^{A}\xi|_{\infty},|N_{n}^{A}\xi|_{\infty}^{-1/|\Gamma|}\big{\\}},\quad\text{for
each }n\in\mathbb{N}.$

In the $\Theta_{t}=\Phi_{t}$ case, this follows from the mean value theorem
and the standard van der Corput lemma. In the $\Theta_{t}=\Psi_{t}$ case, this
follows from the cancellation condition (1.5) and [25, Proposition B.2] (see
[25, p. 21] for details).

  3. Property 3.

For each $\alpha>0$, $n\in\mathbb{N}$, and $\xi\in\mathbb{T}^{\Gamma}$
satisfying

$\bigg{|}\xi_{\gamma}-\frac{a_{\gamma}}{q}\bigg{|}\leq
N_{n}^{-\lvert\gamma\rvert}L\qquad\text{for all }\gamma\in\Gamma$

with $1\leq q\leq L$, $a\in A_{q}^{\Gamma}$, and $1\leq
L\leq\exp\big{(}c\sqrt{\log{N_{n}}}\big{)}(\log N_{n})^{-\alpha}$, we have

$\mathfrak{y}_{N_{n}}(\xi)-\mathfrak{y}_{N_{n-1}}(\xi)=G(a/q)\big{(}\Theta_{N_{n}}(\xi-a/q)-\Theta_{N_{n-1}}(\xi-a/q)\big{)}+\mathcal{O}\big{(}(\log
N_{n})^{-\alpha}\big{)},$

for some constant $c>0$ which is independent of $n,\xi,a$ and $q$.

In the $\mathfrak{y}_{t}=\mathfrak{m}_{t}$, $\Theta_{t}=\Phi_{t}$ case, this
is [38, Property 6]. In the $\mathfrak{y}_{t}=\mathfrak{n}_{t}$,
$\Theta_{t}=\Psi_{t}$ case, this follows from Property 1 alongside Lemma 7
with $\Omega:=\Omega_{N_{n}}\setminus\Omega_{N_{n-1}}$ and
$\mathcal{K}(n,p):=K(n,p)\mathds{1}_{{\Omega}}$, noting the size condition
(1.4) and the continuity condition (1.6). For details see [38, Lemmas 3 and
5].

###  3.3. Parameters discussion

Let $p\in(1,\infty)$ be fixed and let $\chi\in(0,1/10)$. Fix $\tau$ with
$0<\tau<1-\min(2,p)^{-1}$ and let $N_{n}:=\lfloor 2^{n^{\tau}}\rfloor$ for
$n\in\mathbb{N}$. If $p\in(1,2)$, fix $p_{0}$ such that $1<p_{0}<p$. If
instead $p\in(2,\infty)$, fix $p_{0}>p$. If $p=2$, the discussion is moot
since all the interpolation arguments in the article become unnecessary. We
choose $\rho$ with

$\rho>\frac{1}{\tau}\frac{pp_{0}-2p}{2p_{0}-2p}$

so that interpolation of the estimates

$\|T\|_{\ell^{2}}\lesssim
n^{-\rho\tau}\quad\text{and}\quad\|T\|_{\ell^{p_{0}}}\lesssim 1$

yields

$\|T\|_{\ell^{p}}\lesssim n^{-(1+\varepsilon)}\text{ for some }\varepsilon>0.$

Property 1 gives us a corresponding $\beta_{\rho}$. We fix a choice of
$\beta>\beta_{\rho}$ and then fix a choice of $u\in\mathbb{N}$ with
$u>|\Gamma|\beta$. We also have the value of $\delta$ coming from the Gaussian
sum estimate (3.5). With these fixed, we choose the value of $\varrho$ in
Theorem 5 to be

$\varrho:=\min\bigg{(}\frac{\chi}{10u},\frac{\delta}{8\tau}\bigg{)}.$

##  4\. Proof of Theorem 4

By the monotone convergence theorem and standard density arguments it is
enough to prove that

$\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}(M_{t}^{k^{\prime},k^{\prime\prime}}f:t\in\mathbb{I})\lesssim_{p,k,|\Gamma|}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

holds for every finite subset $\mathbb{I}\subset\mathbb{R}_{+}$ with the
implicit constant independent of the set $\mathbb{I}$. We start by splitting
(cf. [16, Lemma 1.3]) into long oscillations/jumps and short variations along
the subexponential sequence $N_{n}$:

$\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\big{(}M_{t}^{k^{\prime},k^{\prime\prime}}f:t\in\mathbb{I}\big{)}\lesssim\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}(T_{\mathbb{Z}^{\Gamma}}[\mathfrak{y}_{N_{n}}]f:n\in\mathbb{N}_{0})+\bigg{\|}\Big{(}\sum_{n\in\mathbb{N}_{0}}V^{2}\big{(}M_{t}^{k^{\prime},k^{\prime\prime}}f:t\in[N_{n},N_{n+1})\cap\mathbb{I}\big{)}^{2}\Big{)}^{1/2}\bigg{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

###  4.1. Short variations

By using the arguments from [26, Section 3.1], the estimate for the short
variations will follow from the estimate

(4.1)
$\big{\|}V^{1}(M_{t}^{k^{\prime},k^{\prime\prime}}f:t\in[N_{n},N_{n+1})\cap\mathbb{I})\big{\|}_{\ell^{1}(\mathbb{Z}^{\Gamma})}\lesssim
n^{\tau-1}\|f\|_{\ell^{1}(\mathbb{Z}^{\Gamma})}.$

Let $t_{1}<t_{2}<\cdots<t_{J(n)}$ be a sequence of elements of
$[N_{n},N_{n+1})\cap\mathbb{I}$. Since the number of elements in
$[N_{n},N_{n+1})\cap\mathbb{I}$ is finite, it is easy to see that

$\big{\|}V^{1}(M_{t}^{k^{\prime},k^{\prime\prime}}f:t\in[N_{n},N_{n+1})\cap\mathbb{I})\big{\|}_{\ell^{1}(\mathbb{Z}^{\Gamma})}\leq\Big{\|}\sum_{j=1}^{J(n)}\big{|}M_{t_{j}}^{k^{\prime},k^{\prime\prime}}f-M_{t_{j-1}}^{k^{\prime},k^{\prime\prime}}f\big{|}\Big{\|}_{\ell^{1}(\mathbb{Z}^{\Gamma})}$

for any $n\in\mathbb{N}_{0}.$ Moreover, we have

(4.2)
$\Big{\|}\sum_{j=1}^{J(n)}\big{|}M_{t_{j}}^{k^{\prime},k^{\prime\prime}}f-M_{t_{j-1}}^{k^{\prime},k^{\prime\prime}}f\big{|}\Big{\|}_{\ell^{1}(\mathbb{Z}^{\Gamma})}\lesssim
2^{-kn^{\tau}}\big{(}\vartheta_{\Omega}(N_{n+1})-\vartheta_{\Omega}(N_{n})\big{)}\|f\|_{\ell^{1}(\mathbb{Z}^{\Gamma})}.$

This follows from the monotonicity of the sets $\Omega_{t}$ and having
$\vartheta_{\Omega}(t)\approx t^{k}$ by the prime number theorem in the
$M_{t}^{k^{\prime},k^{\prime\prime}}=A_{t}^{\mathcal{Q},k^{\prime},k^{\prime\prime}}$
case or the size condition (1.4) in the
$M_{t}^{k^{\prime},k^{\prime\prime}}=H_{t}^{\mathcal{Q},k^{\prime},k^{\prime\prime}}$
case. By [38, Eq. 4.10], the right hand side of (4.2) is bounded by
$n^{\tau-1}\|f\|_{\ell^{1}(\mathbb{Z}^{\Gamma})}$, proving (4.1).

###  4.2. Long oscillations/jumps and the circle method

Let $\eta\colon\mathbb{R}^{\Gamma}\rightarrow[0,1]$ be a smooth function with

$\eta(x)=\begin{cases}1&\text{if }|x|_{\infty}\leq\tfrac{1}{32|\Gamma|},\\\
0&\text{if }|x|_{\infty}\geq\tfrac{1}{16|\Gamma|}.\end{cases}$

For $N\in\mathbb{R}_{+}$, we define the scaling notation

$\eta_{N}(\xi):=\eta\big{(}2^{N\cdot A-N^{\chi}\cdot\rm{Id}}\xi\big{)}$

where $A$ is the matrix given in (3.14) and $\rm{Id}$ is the
$|\Gamma|\times|\Gamma|$ identity matrix. For dyadic integers $s\in
2^{u\mathbb{N}}$, we define the annuli sets of fractions by

(4.3) $\Sigma_{s}:=\begin{cases}\Sigma_{\leq s}&\text{ if }s=2^{u},\\\
\Sigma_{\leq s}\setminus\Sigma_{\leq s/2^{u}}&\text{ if }s>2^{u},\end{cases}$

where the $\Sigma_{\leq\cdot}$ are the sets of Ionescu–Wainger fractions as in
Theorem 5. For $t\geq 2^{u}$, we set $F(t):=\max\\{s\in 2^{u\mathbb{N}}:s\leq
t\\}$. We define

$\Xi_{\leq j^{\tau u}}(\xi):=\sum_{a/q\in\Sigma_{\leq F(j^{\tau
u})}}\eta_{j^{\tau}}(\xi-a/q)$

and, for $s\in 2^{u\mathbb{N}}$, we define the annuli functions

(4.4) $\Xi_{j}^{s}(\xi):=\sum_{a/q\in\Sigma_{s}}\eta_{j^{\tau}}(\xi-a/q).$

By (4.3), we have the telescoping property

$\Xi_{\leq j^{\tau u}}=\sum_{\begin{subarray}{c}s\in 2^{u\mathbb{N}}\\\ s\leq
j^{\tau u}\end{subarray}}\Xi_{j}^{s}.$

Note that $\eta_{j^{\tau}}(\xi)$ satisfies the hypothesis about the support
for $\mathfrak{m}$ in Theorem 5 since
$\frac{1}{8|\Gamma|}2^{-j^{\tau}+j^{\tau\chi}}\leq e^{-j^{2\tau u\varrho}}$
provided that $\varrho\leq\chi/(10u)$. Using the $\Xi_{\leq j^{\tau u}}$
functions, we bound the long oscillations/jumps by

$\displaystyle\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}\sum_{j=1}^{n}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{\leq
j^{\tau
u}}]f:n\in\mathbb{N}\Big{)}+\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}\sum_{j=1}^{n}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})(1-\Xi_{\leq
j^{\tau u}})]f:n\in\mathbb{N}\Big{)}.$

These terms correspond to major and minor arcs respectively.

###  4.3. Minor arcs

Since $V_{1}$ controls the oscillation/jump seminorms, we have

$\displaystyle\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}\sum_{j=1}^{n}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})(1-\Xi_{\leq
j^{\tau
u}})]f:n\in\mathbb{N}\Big{)}\leq\sum_{n=1}^{\infty}\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})(1-\Xi_{\leq
n^{\tau u}})]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

It then suffices to show that

$\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})(1-\Xi_{\leq
n^{\tau u}})]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
n^{-(1+\varepsilon)}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

for some $\varepsilon>0$. This uses Property 1 and follows from the proof of
[38, Eqs. (5.8), (5.9)] with only small changes due to our differing scaling
in the definition of $\eta_{N}(\xi)$. We omit the details.

###  4.4. Introduction to major arcs

Using the annuli multipliers (4.4) and Proposition (3), we bound the major
arcs term by

$\displaystyle\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}\sum_{j=1}^{n}\sum_{\begin{subarray}{c}s\in
2^{u\mathbb{N}}\\\ s\leq j^{\tau
u}\end{subarray}}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{j}^{s}]f:n\in\mathbb{N}\Big{)}\leq\sum_{s\in
2^{u\mathbb{N}}}\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ j\geq s^{1/(\tau
u)}\end{subarray}}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{j}^{s}]f:n\geq
s^{1/\tau u}\Big{)}.$

It then suffices to show for large $s\in 2^{u\mathbb{N}}$ that

(4.5)
$\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\bigg{(}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ j\geq s^{1/(\tau
u)}\end{subarray}}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{j}^{s}]f:n\geq
s^{1/\tau u}\bigg{)}\lesssim
s^{-\varepsilon}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

for some $\varepsilon>0$ since $\sum_{s\in
2^{u\mathbb{N}}}s^{-\varepsilon}<\infty$. Let
$\kappa_{s}:=s^{2\lfloor\varrho\rfloor}$. By splitting the left hand side of
(4.5) at $n\approx 2^{\kappa_{s}}$ into small and large scales, it suffices to
prove that

(4.6)
$\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ j\geq s^{1/(\tau
u)}\end{subarray}}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{j}^{s}]f:n^{\tau}\in[s^{1/u},2^{\kappa_{s}+1}]\Big{)}\lesssim
s^{-\varepsilon}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

and

(4.7)
$\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ j\geq
2^{\kappa_{s}/\tau}\end{subarray}}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{j}^{s}]f:n^{\tau}>2^{\kappa_{s}}\Big{)}\lesssim
s^{-\varepsilon}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

For the small scales (4.6), we will use the Rademacher–Menshov inequality
(2.5) and Theorem 5. For the large scales (4.7), we will use the
Magyar–Stein–Wainger sampling principle from [18, Proposition 2.1] and its
counterpart for the jump inequality from [24, Theorem 1.7]. We first establish
an approximation lemma to replace our discrete multipliers with continuous
counterparts. Let

(4.8)
$v_{j}^{s}(\xi):=\sum_{a/q\in\Sigma_{s}}G(a/q)\big{(}\Theta_{N_{j}}-\Theta_{N_{j-1}}\big{)}(\xi-a/q)\eta_{j^{\tau}}(\xi-a/q)$

and

(4.9)
$\Lambda_{j}^{s}(\xi):=\sum_{a/q\in\Sigma_{s}}\big{(}\Theta_{N_{j}}-\Theta_{N_{j-1}}\big{)}(\xi-a/q)\eta_{j^{\tau}}(\xi-a/q).$

######  Lemma 8.

Let $M\in\mathbb{N}$, $\alpha^{\prime}>0$, and $S_{M}:=\lfloor
2^{M^{\tau}-3M^{\tau\chi}}\rfloor$. For $j\in\mathbb{N}$ with $s^{1/(\tau
u)}\leq j$ and $M\leq j\leq 2M$, we have

(4.10)
$\|(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{j}^{s}-v_{j}^{s}\|_{\ell^{\infty}(\mathbb{T}^{\Gamma})}\lesssim
j^{-\alpha^{\prime}\tau}$

and

(4.11)
$\|(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{j}^{s}-\Lambda_{j}^{s}\mathfrak{m}_{S_{M}}\|_{\ell^{\infty}(\mathbb{T}^{\Gamma})}\lesssim
j^{-\alpha^{\prime}\tau}.$

###### Proof.

For (4.10), since the $\eta_{j^{\tau}}(\xi-a/q)$ bump functions in the
definitions of $\Xi_{j}^{s}$ and $v_{j}^{s}$ have disjoint supports for
distinct fractions $a/q$, it suffices to prove for a fixed $a/q\in\Sigma_{s}$
that

$\big{\|}\eta_{j^{\tau}}(\xi-a/q)\big{[}\big{(}\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}}\big{)}(\xi)-G(a/q)\big{(}\Theta_{N_{j}}-\Theta_{N_{j-1}}\big{)}(\xi-a/q)\big{]}\big{\|}_{\ell^{\infty}(\mathbb{T}^{\Gamma})}\lesssim
j^{-\alpha^{\prime}\tau}$

with the implied constant independent of the choice of $a/q$. Using the
definition of $\Sigma_{s}$, property (i) from Theorem 5, $s\leq j^{\tau u}$,
and $\varrho\leq\chi/(10u)$, we have $q\leq e^{s^{\varrho}}\leq e^{j^{\tau
u\rho}}\leq 2^{j^{\tau\chi}}=:L_{1}.$ On the support of
$\eta_{j^{\tau}}(\xi-a/q)$, we have $|\xi_{\gamma}-a_{\gamma}/q|\lesssim
N_{j}^{-|\gamma|}L_{1}$ for all $\gamma\in\Gamma$. Moreover, we have

$L_{1}=2^{j^{\tau\chi}}\leq
2^{j^{\tau/2}}j^{-\tau\alpha^{\prime}}\lesssim\exp(\sqrt{\log N_{j}})(\log
N_{j})^{-\alpha^{\prime}}.$

The estimate (4.10) then follows from Property 3 with $\alpha=\alpha^{\prime}$
and $L=L_{1}$.

For (4.11), we use (4.10) and are reduced to showing that

$\|v_{j}^{s}-\Lambda_{j}^{s}\mathfrak{m}_{S_{M}}\|_{\ell^{\infty}(\mathbb{T}^{\Gamma})}\lesssim
j^{-\alpha^{\prime}\tau}.$

Fixing $\xi$ in the support of $\eta_{j^{\tau}}(\xi-a/q)$, we have

$\displaystyle|\xi_{\gamma}-a_{\gamma}/q|\leq
2^{-M^{\tau}|\gamma|}2^{(2M)^{\tau\chi}}\leq
2^{-M^{\tau}|\gamma|}2^{2M^{\tau\chi}}=2^{-M^{\tau}|\gamma|}2^{3M^{\tau\chi}|\gamma|}2^{-3M^{\tau\chi}|\gamma|}2^{2M^{\tau\chi}}\leq
S_{M}^{-|\gamma|}2^{-(j/2)^{\tau\chi}}$

for all $\gamma\in\Gamma$. By the triangle inequality, we have

$|G(a/q)-\mathfrak{m}_{S_{M}}(\xi)|\leq|G(a/q)-G(a/q)\Phi_{S_{M}}(\xi-a/q)|+|G(a/q)\Phi_{S_{M}}(\xi-a/q)-\mathfrak{m}_{S_{M}}(\xi)|.$

For the first term, we use the estimate (3.5), the mean value theorem, and
Property 2 to obtain

$\displaystyle|G(a/q)-G(a/q)\Phi_{S_{M}}(\xi-a/q)|\leq
q^{-\delta}\big{|}S_{M}^{A}(\xi-a/q)\big{|}_{\infty}\leq
q^{-\delta}2^{-(j/2)^{\tau\chi}}\lesssim j^{-\alpha^{\prime}\tau}.$

For the second term, we use that

$\displaystyle|\xi_{\gamma}-a_{\gamma}/q|\leq
S_{M}^{-|\gamma|}2^{-M^{\tau\chi}}\leq
S_{M}^{-|\gamma|}2^{(2M)^{\tau\chi}}=:S_{M}^{-|\gamma|}L_{2}$

and $q\leq 2^{j^{\tau\chi}}\leq
2^{(2M)^{\tau\chi}}=L_{2}\leq\exp\big{(}\sqrt{\log S_{M}}\big{)}(\log
S_{M})^{-\alpha^{\prime}/\chi}.$ Hence, we may apply [38, Lemma 3] with
$\alpha=\alpha^{\prime}/\chi$, $N=S_{M}$, and $L=L_{2}$ to obtain

$\displaystyle|G(a/q)\Phi_{S_{M}}(\xi-a/q)-\mathfrak{m}_{S_{M}}(\xi)|\lesssim\log(S_{M})^{-\alpha^{\prime}/\chi}\lesssim(M^{\tau}-3M^{\tau\chi})^{-\alpha^{\prime}/\chi}\lesssim(2M)^{-\alpha^{\prime}\tau}\lesssim
j^{-\alpha^{\prime}\tau}.$

This completes the proof of (4.11). ∎

###  4.5. Small scales

Using that $V_{2}$ dominates oscillations/jumps, splitting
$[s^{1/u},2^{\kappa_{s}+1}]$ into dyadic intervals, and preparing via the
triangle inequality to use (4.11), we bound the left hand side of (4.6) by

$\displaystyle\overbrace{\sum_{M\in
2^{\mathbb{N}}\cap[s^{1/u},2^{\kappa_{s}}]}\Big{\|}V_{2}\Big{(}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ j\geq s^{1/(\tau
u)}\end{subarray}}T_{\mathbb{Z}^{\Gamma}}[\Lambda_{j}^{s}\mathfrak{m}_{S_{M}}]f:n^{\tau}\in[M,2M]\Big{)}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}}^{\text{Main
Term 1}}$ $\displaystyle+\underbrace{\sum_{M\in
2^{\mathbb{N}}\cap[s^{1/u},2^{\kappa_{s}}]}\Big{\|}V_{2}\Big{(}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ j\geq s^{1/(\tau
u)}\end{subarray}}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{j}}-\mathfrak{y}_{N_{j-1}})\Xi_{j}^{s}-\Lambda_{j}^{s}\mathfrak{m}_{S_{M}}]f:n^{\tau}\in[M,2M]\Big{)}\Big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}}_{\text{Error
Term 1}}.$

For Error Term 1, it will suffice to show that

(4.12)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})\Xi_{n}^{s}-\Lambda_{n}^{s}\mathfrak{m}_{S_{M}}]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
n^{-(1+\varepsilon^{\prime})}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

for some $\varepsilon^{\prime}>0$ since we would then bound it by

$\displaystyle\sum_{n\geq s^{1/(\tau
u)}}n^{-(1+\varepsilon^{\prime})}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
s^{-\varepsilon^{\prime}/(\tau
u)}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
s^{-\varepsilon}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

using that $V_{1}$ dominates $V_{2}$. We note by Theorem 5 that

$\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})\Xi_{n}^{s}-\Lambda_{n}^{s}\mathfrak{m}_{S_{M}}]f\big{\|}_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}\lesssim\|f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}$

and, by (4.11) with $\alpha^{\prime}=\rho$, that

$\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})\Xi_{n}^{s}-\Lambda_{n}^{s}\mathfrak{m}_{S_{M}}]f\big{\|}_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim
n^{-\rho\tau}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}.$

Interpolation of the above inequalities yields (4.12).

For Main Term 1, we apply the Rademacher–Menshov inequality (2.5) to bound it
by

$\displaystyle\sum_{M\in
2^{\mathbb{N}}\cap[s^{1/u},2^{\kappa_{s}}]}\sum_{i=0}^{\log_{2}(2M)}\bigg{\|}\Big{(}\sum_{j}\Big{|}\sum_{k\in
I_{i,j}^{M}}T_{\mathbb{Z}^{\Gamma}}[\Lambda_{k}^{s}\mathfrak{m}_{S_{M}}]f\Big{|}^{2}\Big{)}^{1/2}\bigg{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})},$

where $j$ is taken over $j\geq 0$ such that
$I_{i,j}^{M}:=[j2^{i},(j+1)2^{i}]\cap[M^{1/\tau},(2M)^{1/\tau}]\neq\emptyset$.
Let $\tilde{\eta}_{N}(\xi):=\eta_{N}(\xi/2)$. Then
$\tilde{\eta}_{N}\eta_{k^{\tau}}=\eta_{k^{\tau}}$ for $k^{\tau}\geq N$ due to
the nesting supports. This lets us write

$\Lambda_{k}^{s}\mathfrak{m}_{S_{M}}=\Lambda_{k}^{s}\mathfrak{m}_{S_{M}}\sum_{a/q\in\Sigma_{s}}\tilde{\eta}_{M}(\xi-a/q)=:\Lambda_{k}^{s}\mathfrak{m}_{S_{M}}\tilde{\Xi}_{M^{1/\tau}}^{s}$

for $k\in I_{i,j}^{M}$ since then $k\geq M^{1/\tau}$. We have for any
$p\in(1,\infty)$ that

$\bigg{\|}\Big{(}\sum_{j}\Big{|}\sum_{k\in
I_{i,j}^{M}}T_{\mathbb{Z}^{\Gamma}}[\Lambda_{k}^{s}]g\Big{|}^{2}\Big{)}^{1/2}\bigg{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\|g\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

since, by Theorem 5, the above estimate is a consequence of its continuous
counterpart

$\displaystyle\bigg{\lVert}\Big{(}\sum_{j}\big{|}\sum_{k\in
I_{i,j}^{M}}T_{\mathbb{R}^{\Gamma}}\big{[}(\Theta_{N_{k}}-\Theta_{N_{k-1}})\eta_{k}^{\tau}\big{]}f\big{|}^{2}\Big{)}^{1/2}\bigg{\rVert}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim\lVert
f\rVert_{L^{p}(\mathbb{R}^{\Gamma})}.$

The above square function estimate follows by appealing to Property 2 and
arguments from Littlewood–Paley theory. We refer to [25] for more details, see
also [26, Theorem 4.3, p. 42]. Thus,

(4.13) $\bigg{\|}\Big{(}\sum_{j}\Big{|}\sum_{k\in
I_{i,j}^{M}}T_{\mathbb{Z}^{\Gamma}}[\Lambda_{k}^{s}\mathfrak{m}_{S_{M}}]f\Big{|}^{2}\Big{)}^{1/2}\bigg{\|}_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}\lesssim\big{\|}T_{\mathbb{Z}^{\Gamma}}[\mathfrak{m}_{S_{M}}]f\big{\|}_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}\lesssim\|f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}$

using the uniform $\ell^{p}$-boundedness of the averaging operators. We get an
improved bound on $\ell^{2}$. To do this, we show that

$\big{\|}\mathfrak{m}_{S_{M}}\tilde{\Xi}_{M^{1/\tau}}^{s}\big{\|}_{\ell^{\infty}(\mathbb{T}^{\Gamma})}\lesssim
s^{-\delta}$

for $M\in 2^{\mathbb{N}}\cap[s^{1/u},2^{\kappa_{s}}]$. Since the bump
functions in the sum have disjoint supports, it suffices to prove for a fixed
$a/q\in\Sigma_{s}$ that

$\|\mathfrak{m}_{S_{M}}(\xi)\tilde{\eta}_{M}(\xi-a/q)\|_{\ell^{\infty}(\mathbb{T}^{\Gamma})}\lesssim
s^{-\delta}$

with the implied constant independent of the choice of $a/q$. On the support
of $\tilde{\eta}_{M}(\xi-a/q)$, we have

$|\xi_{\gamma}-a_{\gamma}/q|\leq\frac{1}{8|\Gamma|}2^{-M|\gamma|}2^{M^{\chi}}\leq
2^{-M^{\tau}|\gamma|}2^{2M^{\tau\chi}}.$

We follow the same arguments as in the proof of (4.11), choosing
$\alpha^{\prime}=\delta u/\tau$, to show that

$|G(a/q)-\mathfrak{m}_{S_{M}}(\xi)|\lesssim M^{-\delta u}\leq s^{-\delta}.$

For any $\xi\in\mathbb{T}^{\Gamma}$ and $a/q\in\Sigma_{s}$, we have

$\big{|}\mathfrak{m}_{S_{M}}(\xi)\tilde{\Xi}_{M^{1/\tau}}^{s}(\xi)\big{|}\leq|\mathfrak{m}_{S_{M}}(\xi)|\leq|\mathfrak{m}_{S_{M}}(\xi)-G(a/q)|+|G(a/q)|\lesssim
s^{-\delta}$

using that $|G(a/q)|\lesssim q^{-\delta}\lesssim s^{-\delta}$ since $q\geq
s/2^{u}$ by the construction of $\Sigma_{s}$. Hence,

(4.14) $\bigg{\|}\Big{(}\sum_{j}\Big{|}\sum_{k\in
I_{i,j}^{M}}T_{\mathbb{Z}^{\Gamma}}[\Lambda_{k}^{s}\mathfrak{m}_{S_{M}}]f\Big{|}^{2}\Big{)}^{1/2}\bigg{\|}_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim\big{\|}T_{\mathbb{Z}^{\Gamma}}[\mathfrak{m}_{S_{M}}\tilde{\Xi}_{M_{1/\tau}^{s}}]f\big{\|}_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim
s^{-\delta}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}.$

Interpolation of (4.13) with (4.14) then gives that

$\bigg{\|}\Big{(}\sum_{j}\Big{|}\sum_{k\in
I_{i,j}^{M}}T_{\mathbb{Z}^{\Gamma}}[\Lambda_{k}^{s}\mathfrak{m}_{S_{M}}]f\Big{|}^{2}\Big{)}^{1/2}\bigg{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
s^{-8\varrho}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

since $8\varrho\leq\delta/(\rho\tau)$. Thus, we may dominate Main Term 1 by

$\displaystyle\sum_{M\in
2^{\mathbb{N}}\cap[s^{1/u},2^{\kappa_{s}}]}\sum_{i=0}^{\log_{2}(2M)}s^{-8\varrho}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim\kappa_{s}^{2}s^{-8\varrho}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
s^{-4\varrho}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

since $\kappa_{s}\leq s^{2\varrho}$, concluding the proof of (4.6).

###  4.6. Large scales

Since $V_{1}$ dominates $\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}$, we may bound
the left hand side of (4.7) by

$\displaystyle\overbrace{\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}\sum_{\begin{subarray}{c}1\leq
j\leq n\\\ j\geq
2^{\kappa_{s}/\tau}\end{subarray}}T_{\mathbb{Z}^{\Gamma}}[v_{j}^{s}]f:n^{\tau}>2^{\kappa_{s}}\Big{)}}^{\text{Main
Term 2}}+\overbrace{\sum_{n\geq
2^{\kappa_{s}/\tau}}\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})\Xi_{n}^{s}-v_{n}^{s}]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}}^{\text{Error
Term 2}}.$

For Error Term 2, it will suffice to show that

(4.15)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})\Xi_{n}^{s}-v_{n}^{s}]f\big{\|}_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
e^{(|\Gamma|+1)s^{\varrho}}n^{-(1+\varepsilon^{\prime})}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

for some $\varepsilon^{\prime}>0$ since we would then bound it by

$\displaystyle e^{(|\Gamma|+1)s^{\varrho}}\sum_{n\geq
2^{\kappa_{s}/\tau}}n^{-(1+\varepsilon^{\prime})}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
e^{(|\Gamma|+1)s^{\varrho}}2^{-s^{2\varrho}\varepsilon^{\prime}/\tau}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
s^{-\varepsilon}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}.$

We have

$\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})\Xi_{n}^{s}-v_{n}^{s}]f\big{\|}_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim
n^{-\rho\tau}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}$

by (4.10) with $\alpha^{\prime}=\rho$. We also have

$\big{\|}T_{\mathbb{Z}^{\Gamma}}[(\mathfrak{y}_{N_{n}}-\mathfrak{y}_{N_{n-1}})\Xi_{n}^{s}-v_{n}^{s}]f\big{\|}_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}\lesssim
e^{(|\Gamma|+1)s^{\varrho}}\|f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}$

simply by the triangle inequality and property (i) from Theorem 5.
Consequently (4.15) follows by interpolation.

For Main Term 2, we define

$w^{s}(\xi):=\sum_{a/q\in\Sigma_{s}}G(a/q)\tilde{\eta}_{2^{\kappa_{s}}}(\xi-a/q),\quad\Pi^{s}(\xi):=\sum_{a/q\in\Sigma_{s}}\tilde{\eta}_{2^{\kappa_{s}}}(\xi-a/q),$

and

$\omega_{n}^{s}(\xi):=\sum_{2^{\kappa_{s}/\tau}\leq j\leq
n}(\Theta_{N_{j}}-\Theta_{N_{j-1}})(\xi)\eta_{j^{\tau}}(\xi).$

Let $Q_{s}:=\operatorname{lcm}(q:a/q\in\Sigma_{s}).$ By property (iv) from
Theorem 5, we have $Q_{s}\leq 3^{s}$. The function $\omega_{n}^{s}$ is
supported on $[-\frac{1}{4Q_{s}},\frac{1}{4Q_{s}}]$ for large $s\in
2^{u\mathbb{N}}$ since, on the support of $\eta_{2^{\kappa_{s}}}$, we have
$|\xi_{\gamma}|\leq 2^{-2^{-\kappa_{s}}+2^{\kappa_{s}\chi}}\leq(4Q_{s})^{-1}$
for all $\gamma\in\Gamma$ and large $s$. We also have

$\sum_{2^{\kappa_{s}/\tau}\leq j\leq
n}v_{j}^{s}(\xi)=w^{s}(\xi)\sum_{b\in\mathbb{Z}^{\Gamma}}\omega_{n}^{s}(\xi-b/Q_{s}).$

Therefore, it suffices to prove

(4.16)
$\mathcal{S}_{\mathbb{Z}^{\Gamma}}^{p}\Big{(}T_{\mathbb{Z}^{\Gamma}}\Big{[}\sum_{b\in\mathbb{Z}^{\Gamma}}\omega_{n}^{s}(\cdot-b/Q_{s})\Big{]}f:n^{\tau}>2^{\kappa_{s}}\Big{)}\lesssim\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

and

(4.17)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[w^{s}]f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}\lesssim
s^{-\varepsilon}\|f\|_{\ell^{p}(\mathbb{Z}^{\Gamma})}$

for some $\varepsilon>0$.

By the Magyar–Stein–Wainger sampling principle [18, Proposition 2.1] for the
oscillation seminorm or the sampling principle for the jumps [24, Theorem
1.7], (4.16) follows from

(4.18)
$\mathcal{S}_{\mathbb{R}^{\Gamma}}^{p}(T_{\mathbb{R}^{\Gamma}}[\omega_{n}^{s}]f:n^{\tau}>2^{\kappa_{s}})\lesssim\|f\|_{L^{p}(\mathbb{R}^{\Gamma})}.$

To prove (4.18), we use that the $\omega_{n}^{s}$ functions are almost
telescoping. We define

$\Delta_{n}^{s}(\xi):=\sum_{2^{\kappa_{s}/\tau}\leq j\leq
n}(\Theta_{N_{j}}-\Theta_{N_{j-1}})(\xi)=(\Theta_{N_{n}}-\Theta_{N_{2^{\kappa_{s}/\tau}-1}})(\xi).$

Then (4.18) follows from

(4.19)
$\mathcal{S}_{\mathbb{R}^{\Gamma}}^{p}(T_{\mathbb{R}^{\Gamma}}[\Delta_{n}^{s}]f:n^{\tau}>2^{\kappa_{s}})\lesssim\|f\|_{L^{p}(\mathbb{R}^{\Gamma})}$

since the error term is bounded by

$\sum_{n>2^{\kappa_{s}/\tau}}\big{\|}T_{\mathbb{R}^{\Gamma}}[(\Theta_{N_{n}}-\Theta_{N_{n-1}})(\eta_{n^{\tau}}-1)]f\big{\|}_{L^{p}(\mathbb{R}^{\Gamma})}\lesssim\|f\|_{L^{p}(\mathbb{R}^{\Gamma})}$

using Property 2 and interpolation.

On the other hand, due to translation invariance of
$\mathcal{S}_{\mathbb{R}^{\Gamma}}^{p}$, (4.19) follows from

(4.20)
$\displaystyle\mathcal{S}_{\mathbb{R}^{\Gamma}}^{p}(T_{\mathbb{R}^{\Gamma}}[\Theta_{t}]f:t>0)\lesssim\|f\|_{L^{p}(\mathbb{R}^{\Gamma})}.$

For the jump inequality, the estimate (4.20) was proven in [25, Theorem 1.22,
Theorem 1.30] for both $\Phi_{t}$ and $\Psi_{t}$. For the oscillation
inequality, (4.20) was proven in [20, Eq. 3.38] for $\Phi_{t}$ and in [35,
Theorem 1.9] for $\Psi_{t}$. This concludes the proof of (4.16).

For (4.17), we note by (3.5) that

(4.21)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[w^{s}]f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim
s^{-\delta}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}.$

On $\ell^{p_{0}}$, we start by splitting

$w^{s}=\Pi^{s}\mathfrak{m}_{J_{s}}+(w^{s}-\Pi^{s}\mathfrak{m}_{J_{s}}),$

where $J_{s}=\lfloor 2^{2^{\kappa_{s}}-3\cdot 2^{\kappa_{s}\chi}}\rfloor$. By
Theorem 5, we have

(4.22)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[\Pi^{s}\mathfrak{m}_{J_{s}}]f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}\lesssim\|f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}.$

Let $p_{00}\in(1,\infty)$. Then

(4.23)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[w^{s}-\Pi^{s}\mathfrak{m}_{J_{s}}]f\|_{\ell^{p_{00}}(\mathbb{Z}^{\Gamma})}\lesssim
e^{(|\Gamma|+1)s^{\varrho}}\|f\|_{\ell^{p_{00}}(\mathbb{Z}^{\Gamma})}$

by property (i) from Theorem 5. Therefore, it suffices to show that

(4.24)
$\big{\|}T_{\mathbb{Z}^{\Gamma}}[w^{s}-\Pi^{s}\mathfrak{m}_{J_{s}}]f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}\lesssim
2^{-\chi s^{2\rho}}\|f\|_{\ell^{2}(\mathbb{Z}^{\Gamma})}$

since interpolating (4.24) with (4.23) for an appropriate choice of $p_{00}$
gives

$\big{\|}T_{\mathbb{Z}^{\Gamma}}[w^{s}-\Pi^{s}\mathfrak{m}_{J_{s}}]f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}\lesssim\|f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})},$

combining this with (4.22) gives

$\big{\|}T_{\mathbb{Z}^{\Gamma}}[w^{s}]f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})}\lesssim\|f\|_{\ell^{p_{0}}(\mathbb{Z}^{\Gamma})},$

and interpolating the above inequality with (4.21) completes the proof of
(4.17) and, thereby, that of (4.7). The proof of (4.24) will proceed similarly
as in the proof of (4.11). For $\xi$ in the support of
$\tilde{\eta}_{2^{\kappa_{s}}}(\xi-a/q)$, we have

$\displaystyle|\xi_{\gamma}-a_{\gamma}/q|\leq\frac{1}{16|\Gamma|}2^{-2^{\kappa_{s}}|\gamma|}2^{2^{\kappa_{s}\chi}}\leq
2^{-2^{\kappa_{s}}|\gamma|}2^{2^{\kappa_{s}\chi}}\leq
J_{s}^{-|\gamma|}2^{-2^{\kappa_{s}\chi}}$

for all $\gamma\in\Gamma$. By the triangle inequality, we have

$|G(a/q)-\mathfrak{m}_{J_{s}}(\xi)|\leq|G(a/q)-G(a/q)\Phi_{J_{s}}(\xi-a/q)|+|G(a/q)\Phi_{J_{s}}(\xi-a/q)-\mathfrak{m}_{J_{s}}(\xi)|.$

For the first term, we use (3.5) and the mean value theorem to write

$\displaystyle|G(a/q)-G(a/q)\Phi_{J_{s}}(\xi-a/q)|\leq
q^{-\delta}\big{|}J_{s}^{A}(\xi-a/q)\big{|}_{\infty}\leq
q^{-\delta}2^{-2^{\kappa_{s}\chi}}\lesssim 2^{-\chi s^{2\varrho}}.$

For the second term, we use that

$\displaystyle|\xi_{\gamma}-a_{\gamma}/q|\leq
J_{s}^{-|\gamma|}2^{-2^{\kappa_{s}\chi}}\leq
J_{s}^{-|\gamma|}e^{s^{\varrho}}=:J_{s}^{-|\gamma|}L_{3}$

and $q\leq L_{3}\leq\exp\big{(}\sqrt{\log J_{s}}\big{)}(\log J_{s})^{-1}$ to
apply [38, Lemma 3] with $\alpha=1$, $N=J_{s}$, and $L=L_{3}$. This gives

$\displaystyle|G(a/q)\Phi_{J_{s}}(\xi-a/q)-\mathfrak{m}_{J_{s}}(\xi)|\lesssim\log(J_{s})^{-1}\lesssim(2^{\kappa_{s}}-3\cdot
2^{\kappa_{s}\chi})^{-1}\lesssim 2^{-\chi\kappa_{s}}\lesssim 2^{-\chi
s^{2\varrho}},$

completing the proof.

##  5\. Remarks

As a simple consequence of our results, we can prove the convergence of the
Wiener–Wintner type averages. This result is probably known, but we have not
found anything like this in the literature in the presented generality.

Let $(X,\mu)$ be a measure space endowed with a measure preserving
transformation $T\colon X\to X$ and let

$R(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots+a_{1}x+a_{0},\quad x\in\mathbb{R},$

be a polynomial with real coefficients. Moreover, let
$P\colon\mathbb{Z}\to\mathbb{Z}$ be a polynomial with integer coefficients
such that $P(0)=0$. For $p\in(1,\infty)$, the Wiener–Wintner type averages

(5.1) $\frac{1}{2N+1}\sum_{n=-N}^{N}f(T^{P(n)}x)\bm{e}\big{(}R(n)\big{)}$

converge $\mu$-almost everywhere for any $f\in L^{p}(X,\mu)$. According to
Assani [1, p. 179], the convergence of the averages (5.1) in the case when
$\deg P\geq 2$ is known only for $R\equiv 0$. However, in [9, Theorem 1.9],
the authors have established the convergence in the case when $P(n)=n^{2}$ and
$R(n)=\theta n$ for any $\theta\in\mathbb{R}.$

Let us show how to deduce the convergence of the averages (5.1) from Corollary
2. Clearly, we may assume that $R(0)=0$. We consider the measure space
$(Y,\nu)$ where $Y:=X\times\mathbb{T}$, $\nu:=\mu\times\lambda$, and $\lambda$
is the normalized Lebesgue measure on $\mathbb{T}$. We equip the space
$(Y,\nu)$ with the family of measure preserving commuting transformations
$S_{1},S_{2},\ldots,S_{m},S_{m+1}\colon X\times\mathbb{T}\to
X\times\mathbb{T}$ where, for $j=1,\ldots,m$, we put $S_{j}:={\rm Id}\times
D_{j}$ with

$D_{j}(\xi):=\bm{e}(a_{j})\xi$

being a rotation on $\mathbb{T}$, and $S_{m+1}:=T\times{\rm Id}.$ We consider
the following polynomial mapping

$\mathcal{P}(n):=(n^{1},n^{2},\ldots,n^{m},P(n))\colon\mathbb{Z}\to\mathbb{Z}^{m+1}.$

By Corollary 2, we know that the averages

(5.2) $\frac{1}{2N+1}\sum_{n=-N}^{N}h\big{(}S_{1}^{n^{1}}\cdots
S_{m}^{n^{m}}S_{m+1}^{P(n)}y\big{)},\quad y\in X\times\mathbb{T}$

converge $\nu$-almost everywhere for any $h\in L^{p}(Y,\nu)$. If, for $f\in
L^{p}(X,\mu)$, we consider the function $h(y):=f(x)\xi$, then we see that the
convergence of the averages (5.1) follows from the convergence of the averages
(5.2).

The procedure described above can be extended to obtain that, for
$\mathcal{P}$ being a polynomial mapping of the form (1.1) and
$\mathcal{R}\colon\mathbb{R}^{k}\to\mathbb{R}$ being a polynomial with real
coefficients, the averages

$\mathcal{A}_{t}^{\mathcal{P},\mathcal{R},k^{\prime},k^{\prime\prime}}f(x):=\frac{1}{\vartheta_{\Omega}(t)}\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}f(S_{1}^{\mathcal{P}_{1}(n,p)}\cdots
S_{d}^{\mathcal{P}_{d}(n,p)}x)\mathds{1}_{\Omega_{t}}(n,p)\bm{e}\big{(}\mathcal{R}(n,p)\big{)}\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)}$

and

$\mathcal{H}_{t}^{\mathcal{P},\mathcal{R},k^{\prime},k^{\prime\prime}}f(x):=\sum_{(n,p)\in\mathbb{Z}^{k^{\prime}}\times(\pm\mathbb{P})^{k^{\prime\prime}}}f(S_{1}^{\mathcal{P}_{1}(n,p)}\cdots
S_{d}^{\mathcal{P}_{d}(n,p)}x)K(n,p)\mathds{1}_{\Omega_{t}}(n,p)\bm{e}\big{(}\mathcal{R}(n,p)\big{)}\Big{(}\prod_{i=1}^{k^{\prime\prime}}\log|p_{i}|\Big{)}$

converge $\mu$-almost everywhere for any $f\in L^{p}(X,\mu)$ with
$p\in(1,\infty)$. Moreover, we can deduce that the analogue of Corollary 2
holds for
$\mathcal{A}_{t}^{\mathcal{P},\mathcal{R},k^{\prime},k^{\prime\prime}}$ and
$\mathcal{H}_{t}^{\mathcal{P},\mathcal{R},k^{\prime},k^{\prime\prime}}$.

Unfortunately, we are not able to prove the Wiener–Wintner theorem for the
averages
$\mathcal{A}_{t}^{\mathcal{P},\mathcal{R},k^{\prime},k^{\prime\prime}}$ and
$\mathcal{H}_{t}^{\mathcal{P},\mathcal{R},k^{\prime},k^{\prime\prime}}$. In
our case, that would mean showing that, for any $M\in\mathbb{N}$, there is a
subset of $X$ of full measure on which the convergence holds regardless of the
choice of polynomial $\mathcal{R}$ with $\deg\mathcal{R}\leq M$.

It is an interesting question whenever the Wiener–Wintner theorem can be
somehow deduced from the inequality

(5.3)
$\sup_{N\in\mathbb{N}}\sup_{I\in\mathfrak{S}_{N}(\mathbb{R}_{+})}\big{\lVert}O_{I,N}^{2}(\mathcal{A}_{t}^{\mathcal{P},\mathcal{R},k^{\prime},k^{\prime\prime}}f:t>0)\big{\rVert}_{L^{p}(X,\mu)}\leq
C_{p,d,k,\deg\mathcal{P},\deg\mathcal{R}}\lVert f\rVert_{L^{p}(X,\mu)}.$

This question is motivated by the fact that the constant in (5.3) depends only
on the degree of $\mathcal{R}$ and not its coefficients. We hope to
investigate this problem in the near future.

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