arXiv subjects

Mariusz Mirek

Publications and source records attributed to Mariusz Mirek.

At least 19 recordsLinked to original sources

arXiv2026

Discrete analogues in harmonic analysis: Multi-parameter Radon averages

In this paper we study maximal and oscillation inequalities for multi-parameter discrete Radon averaging operators. We develop a robust variant of the multi-parameter circle method within the framework of Discrete Analogues in Harmonic Analysis. In particular, this gives quantitative estimates for these averages and their underlying Fourier multipliers which reveals an interesting major-arcs rigidity phenomenon. As a consequence, we completely resolve in the affirmative the multi-parameter Bellow--Furstenberg problem in pointwise ergodic theory.

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arXiv2026

A remark on Kolmogorov's theorem

The aim of this note is to illustrate that the set of integrable functions on the torus $\mathbb{T}$ for which the Fourier series diverges almost everywhere is of the second Baire category in $L^1(\mathbb{T})$.

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arXiv2026

Remarks on the Ionescu-Wainger multiplier theorem

In this paper, we extend the recent Ionescu--Wainger multiplier theorem for the set of canonical fractions by Kosz, Mirek, Peluse, Wan, and Wright in several directions. First, we prove its weighted version, which allows us to combine a multifrequency setting with appropriate arithmetic weights. Second, we establish useful seminorm variants of the theorem. Third, we improve the norm upper bounds and, surprisingly, show that these bounds cannot be uniform in the size of the family of canonical fractions. Finally, we demonstrate how these refinements (especially handling arithmetic weights) can be applied by giving a short proof of Bourgain's pointwise ergodic theorem for polynomial iterates.

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arXiv2026

Discrete analogues in harmonic analysis: $TT^*$ methods

In this note we present how the almost-orthogonality methods based on $TT^*$ arguments can be employed to study boundedness of discrete operators of Radon type. Almost-orthogonality methods have particular significance when the classical Fourier methods are not available. However here, to avoid technicalities and present the key ideas behind the discrete almost-orthogonality methods, we give a new proof of the $\ell^2(\mathbb{Z}^d)$-boundedness of Bourgain's maximal inequality for Radon polynomial averages.

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arXiv2026

The circle method and pointwise ergodic theorems

The purpose of this article is to discuss the circle method and its quantitative role in understanding pointwise almost everywhere convergence phenomena for polynomial ergodic averaging operators. Specifically, we will use the circle method to illustrate that pointwise almost everywhere convergence and norm convergence in ergodic theory can have fundamentally different natures. More importantly, these differences may necessitate the use of distinct types of tools, which can sometimes be more intriguing than the original problems themselves.

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arXiv2026

Polynomial ergodic theorems in the spirit of Dunford and Zygmund

The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of uniform oscillation estimates for multiparameter polynomial Radon averaging operators. In the proof of our main result we develop flexible Fourier methods that exhibit and handle the so-called "parameters-gluing'' phenomenon, an obstruction that arises in studying oscillation and variation inequalities for multiparameter polynomial Radon operators. We will also discuss connections of our main result with a multiparameter variant of the Bellow-Furstenberg problem.

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arXiv2025

Pointwise convergence of polynomial multiple ergodic averages along the primes

We establish pointwise almost everywhere convergence for the polynomial multilinear ergodic averages $$\frac{1}{N} \sum_{n=1}^N \La(n) f_1(T^{P_1(n)} x)\cdots f_k(T^{P_k(n)} x)$$ as $N\to \infty$, where $\La$ is the von Mangoldt function, $T \colon X \to X$ is an invertible measure-preserving transformation of a probability space $(X,\nu)$, $P_1,\ldots, P_k$ are polynomials with integer coefficients and distinct degrees, and $f_1,\ldots,f_k\in L^\infty(X)$. This pointwise almost everywhere convergence result can be seen as a refinement of the norm convergence result obtained in Wooley--Ziegler (Amer. J. Math, 2012) in the case of polynomials with distinct degrees. We develop a multilinear circle method for von Mangoldt-weighted (equivalently, prime-weighted) averages in the general $k$-linear setting. The advantage of our method, besides establishing Weyl-type inequalities for multilinear Cram{\'e}r-weighted averages and sharp $p$-adic $L^q$-improving multilinear estimates among other tools, is that for the first time it allows us to work with inverse theorems having subpolynomial bounds in the general multilinear setting. This, in turn, yields sharp $r$-variational estimates $r > 2$ for our weighted polynomial multilinear ergodic average and, more importantly, offers prospects for addressing other multilinear problems involving inverse theorems lacking polynomial bounds.

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arXiv2025

Dimension-free estimates for discrete maximal functions related to normalized gaussians

In this paper, we investigate dimension-free estimates for maximal operators of convolutions with discrete normalized Gaussians (related to the Theta function) in the context of maximal, jump and $r$-variational inequalities on $\ell^p(\mathbb{Z}^d)$ spaces. This is the first instance of a discrete operator in the literature where $\ell^p(\mathbb{Z}^d)$ bounds are provided for the entire range of $1 < p < \infty$. The methods of proof rely on developing robust Fourier methods, which are combined with the fractional derivative, a tool that has not been previously applied to studying similar questions in the discrete setting.

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arXiv2024

The multilinear circle method and a question of Bergelson

Let $k\in \mathbb Z_+$ and $(X, \mathcal B(X), \mu)$ be a probability space equipped with a family of commuting invertible measure-preserving transformations $T_1,\ldots, T_k \colon X\to X$. Let $P_1,\ldots, P_k\in\mathbb Z[\rm n]$ be polynomials with integer coefficients and distinct degrees. We establish pointwise almost everywhere convergence of the multilinear polynomial ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf_1\big(T_1^{P_1(n)}x\big)\cdots f_k\big(T_k^{P_k(n)}x\big), \qquad x\in X, \] as $N\to\infty$ for any functions $f_1, \ldots, f_k\in L^{\infty}(X)$. Besides a couple of results in the bilinear setting $k=2$, and then only in the single transformation case $T_1 = T_2$, this is the first pointwise result for general polynomial multilinear ergodic averages in arbitrary measure-preserving systems. This answers a question of Bergelson from 1996 in the affirmative for any polynomials with distinct degrees, and makes progress on the Furstenberg--Bergelson--Leibman conjecture. In this paper, we build a versatile \emph{multilinear circle method} by developing the Ionescu--Wainger multiplier theorem for the set of canonical fractions, which gives a positive answer to a question of Ionescu and Wainger from 2005. We also establish multilinear $L^p$-improving bounds and an inverse theorem in higher order Fourier analysis for averages over polynomial corner configurations, which we use to establish a multilinear analogue of Weyl's inequality and its real counterpart, a Sobolev smoothing estimate.

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arXiv2024

Polynomial progressions in topological fields

Let $P_1, \ldots, P_m \in K[y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y), \ldots, x + P_m(y)$ lying in a set $S\subseteq K$ of positive density. The proof relies on a general $L^{\infty}$ inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex and $p$-adic analysis.

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arXiv2023

On a multi-parameter variant of the Bellow-Furstenberg problem

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow-Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

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arXiv2022

Polynomial sequences in discrete nilpotent groups of step 2

We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular, we present what we call a "nilpotent circle method" that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

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arXiv2022

Some remarks on oscillation inequalities

In this paper we establish uniform oscillation estimates on $L^p(X)$ with $p\in(1,\infty)$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by Rosenblatt and Wierdl in the early 1990's. We also give a slightly different proof of the uniform oscillation inequality of Jones, Kaufman, Rosenblatt and Wierdl for bounded martingales. Finally, we show that oscillations, in contrast to jump inequalities, cannot be seen as an endpoint for $r$-variation inequalities.

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arXiv2022

Polynomial averages and pointwise ergodic theorems on nilpotent groups

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $σ$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $ρ$-variational inequalities on $L^2$ for $2<ρ<\infty$. This gives an affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

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arXiv2022

Oscillation inequalities in ergodic theory and analysis: one-parameter and multi-parameter perspectives

In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.

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arXiv2022

Pointwise ergodic theorems for non-conventional bilinear polynomial averages

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to \infty$, where $T \colon X \to X$ is a measure-preserving transformation of a $σ$-finite measure space $(X,μ)$, $P(\mathrm{n}) \in \mathbb Z[\mathrm{n}]$ is a polynomial of degree $d \geq 2$, and $f \in L^{p_1}(X), \ g \in L^{p_2}(X)$ for some $p_1,p_2 > 1$ with $\frac{1}{p_1} + \frac{1}{p_2} \leq 1$. We also establish an $r$-variational inequality for these averages (at lacunary scales) in the optimal range $r > 2$. We are also able to "break duality" by handling some ranges of exponents $p_1,p_2$ with $\frac{1}{p_1}+\frac{1}{p_2} > 1$, at the cost of increasing $r$ slightly. This gives an affirmative answer to Problem 11 from Frantzikinakis' open problems survey for the Furstenberg--Weiss averages (with $P(\mathrm{n})=\mathrm{n}^2$), which is a bilinear variant of Question 9 considered by Bergelson in his survey on Ergodic Ramsey Theory from 1996. This also gives a contribution to the Furstenberg-Bergelson-Leibman conjecture. Our methods combine techniques from harmonic analysis with the recent inverse theorems of Peluse and Prendiville in additive combinatorics. At large scales, the harmonic analysis of the adelic integers $\mathbb A_{\mathbb Z}$ also plays a role.

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arXiv2021

Some remarks on dimension-free estimates for the discrete Hardy-Littlewood maximal functions

Dependencies of the optimal constants in strong and weak type bounds will be studied between maximal functions corresponding to the Hardy--Littlewood averaging operators over convex symmetric bodies acting on $\mathbb R^d$ and $\mathbb Z^d$. Firstly, we show, in the full range of $p\in[1,\infty]$, that these optimal constants in $L^p(\mathbb R^d)$ are always not larger than their discrete analogues in $\ell^p(\mathbb Z^d)$; and we also show that the equality holds for the cubes in the case of $p=1$. This in particular implies that the best constant in the weak type $(1,1)$ inequality for the discrete Hardy--Littlewood maximal function associated with centered cubes in $\mathbb Z^d$ grows to infinity as $d\to\infty$, and if $d=1$ it is equal to the largest root of the quadratic equation $12C^2-22C+5=0$. Secondly, we prove dimension-free estimates for the $\ell^p(\mathbb Z^d)$ norms, $p\in(1,\infty]$, of the discrete Hardy--Littlewood maximal operators with the restricted range of scales $t\geq C_q d$ corresponding to $q$-balls, $q\in[2,\infty)$. Finally, we extend the latter result on $\ell^2(\mathbb Z^d)$ for the maximal operators restricted to dyadic scales $2^n\ge C_q d^{1/q}$.

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arXiv2021

Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres

We establish an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{h_1, h_2, h_3}(λ): =\{x\in\mathbb Z_+^3:\lfloor h_1(x_1)\rfloor+\lfloor h_2(x_2)\rfloor+\lfloor h_3(x_3)\rfloor=λ\} \quad \text{with}\quad λ\in\mathbb Z_+; \] where functions $h_1, h_2, h_3$ are constant multiples of regularly varying functions of the form $h(x):=x^c\ell_h(x)$, where the exponent $c>1$ (but close to $1$) and a function $\ell_h(x)$ is taken from a certain wide class of slowly varying functions. Taking $h_1(x)=h_2(x)=h_3(x)=x^c$ we will also derive an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{c}^3(λ) := \{x \in \mathbb Z^3 : \lfloor |x_1|^c \rfloor + \lfloor |x_2|^c \rfloor + \lfloor |x_3|^c \rfloor= λ\} \quad \text{with}\quad λ\in\mathbb Z_+; \] which can be thought of as a perturbation of the classical Waring problem in three variables. We will use the latter asymptotic formula to study, the main results of this paper, norm and pointwise convergence of the ergodic averages \[ \frac{1}{\#\mathbf S_{c}^3(λ)}\sum_{n\in \mathbf S_{c}^3(λ)}f(T_1^{n_1}T_2^{n_2}T_3^{n_3}x) \quad \text{as}\quad λ\to\infty; \] where $T_1, T_2, T_3:X\to X$ are commuting invertible and measure-preserving transformations of a $σ$-finite measure space $(X, ν)$ for any function $f\in L^p(X)$ with $p>\frac{11-4c}{11-7c}$. Finally, we will study the equidistribution problem corresponding to the spheres $\mathbf S_{c}^3(λ)$.

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