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Máté Wierdl

Publications and source records attributed to Máté Wierdl.

7 recordsLinked to original sources
arXiv2026

Pointwise ergodic theorems along sequences of intermediate growth

We establish the first pointwise convergence result for ergodic averages with iterates along explicit and deterministic sequences of intermediate growth, that is, growing faster than any polynomial but slower than any exponential. In particular, we show that the sequence $(\lfloor \exp((\log n)^c)\rfloor)_{n\in\mathbb{Z}+}$, with $c\in(1,8/7)$, is universally $L^p$-good for every $p\in(1,\infty]$. This gives an affirmative answer to an open problem dating back to the mid 1980s and contributes to Bellow's program, initiated in the earlier part of the same decade, on the characterization of $L^p$-good sequences in pointwise ergodic theorems. The proof combines the so-called one-frequency circle method with a delicate application of Vinogradov's method for estimating exponential sums whose phases involve $\big(\lfloor \exp((\log n)^c)\rfloor\big)_{n\in\mathbb{Z}_+}$. An interesting feature of our analysis, reminiscent of estimates arising in the study of the zero-free region of the Riemann zeta function, is that the argument relies on the classical Vinogradov method, in the sense that it necessitates estimates on the number of solutions for the Vinogradov system of Diophantine equations with explicit dependence on the system's parameters.

math.NT↗
arXiv2025

Fluctuation of ergodic averages and other stochastic processes

For an ergodic map $T$ and a non-constant, real-valued $f \in L^1$, the ergodic averages $\mathbb{A}_N f(x) = \frac{1} {N} \sum_{n=1}^N f(T^n x)$ converge a.e., but the convergence is never monotone. Depending on particular properties of the function $f$, the averages $\mathbb{A}_N f(x)$ may or may not actually fluctuate around the mean value infinitely often a.e. We will prove that a.e. fluctuation around the mean is the generic behavior. That is, for a fixed ergodic $T$, the generic non-constant $f\in L^1$ has the averages $\mathbb{A}_N f(x)$ fluctuating around the mean infinitely often for almost every $x$. We also consider fluctuation for other stochastic processes like subsequences of the ergodic averages, convolution operators, weighted averages, uniform distribution and martingales. We will show that in general, in these settings fluctuation around the limit infinitely often persists as the generic behavior.

math.DS↗
arXiv2023

Sublacunary sequences that are strong sweeping out

An increasing sequence $(a_n)$ of positive integers which satisfies $\frac{a_{n+1}}{a_n}>1+η$ for some positive $η$ is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence is strong sweeping out which means that in every aperiodic dynamical system we can find a set $E$ of arbitrary small measure so that $\limsup_N\frac{1}{N} \sum_{n\le N}\mathbb{1}_E(T^nx)=1$ and $\liminf_N\frac{1}{N} \sum_{n\le N}\mathbb{1}_E(T^nx)=0$ almost everywhere. In this paper we improve this result by showing that if $(a_n)$ satisfies only $\frac{a_{n+1}}{a_n}>1+\frac1{(\log\log n)^{1-η}}$ for some positive $η$ then it is already strong sweeping out.

math.DS↗
arXiv2017

On Convergence of Oscillatory Ergodic Hilbert Transforms

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove the following ergodic theorem: let $p(t)$ be a Hardy field function which grows "super-linearly" and stays "sufficiently far" from polynomials. We show that for each measure-preserving system, $(X,Σ,μ,τ)$, with $τ$ a measure-preserving $\mathbb{Z}$-action, the modulated one-sided ergodic Hilbert transform \[ \sum_{n=1}^\infty \frac{e^{2πi p(n)}}{n} τ^n f(x) \] converges $μ$-a.e. for each $f \in L^r(X), \ 1 \leq r < \infty$. This affirmatively answers a question of J. Rosenblatt. In the second part of the paper, we establish almost sure sparse bounds for random one-sided ergodic Hilbert transforms, \[ \sum_{n=1}^\infty \frac{X_n}{n} τ^n f(x), \] where $\{ X_n \}$ are uniformly bounded, independent, and mean-zero random variables.

math.CA↗
arXiv2016

Under recurrence in the Khintchine recurrence theorem

The Khintchine recurrence theorem asserts that on a measure preserving system, for every set $A$ and $\varepsilon>0$, we have $μ(A\cap T^{-n}A)\geq μ(A)^2-\varepsilon$ for infinitely many $n\in \mathbb{N}$. We show that there are systems having under-recurrent sets $A$, in the sense that the inequality $μ(A\cap T^{-n}A)< μ(A)^2$ holds for every $n\in \mathbb{N}$. In particular, all ergodic systems of positive entropy have under-recurrent sets. On the other hand, answering a question of V.~Bergelson, we show that not all mixing systems have under-recurrent sets. We also study variants of these problems where the previous strict inequality is reversed, and deduce that under-recurrence is a much more rare phenomenon than over-recurrence. Finally, we study related problems pertaining to multiple recurrence and derive some interesting combinatorial consequences.

math.DS↗
arXiv2016

Slowly decaying averages and fat towers

Let $(X,Σ,m,τ)$ be an ergodic system, that is, $(X, Σ, m)$ is a probability space and $τ: X \to X$ is an invertible ergodic $m$-preserving transformation. For a function $f:X\to\mathbb R$, let $A_Nf$ denote the $N$th ergodic average, $A_Nf(x)=\frac{1}{N}\cdot (f(x)+\dots+τ^ {N-1}f(x))$. Martin Barlow (personal communication) asked the following question, which arose from the work of a student (Zichun Ye) on interface models. Question: If $f(x) \ge 0$ is integrable, and $N(x) = \min \{n: A_kf(x) \le 2 \int f \text{for all} k \ge n\}$, is it the case that $N(x)$ is also integrable? In this note we show that the answer to this Question is no in general, even for bounded functions. In so doing we discover that every ergodic system has a special sort of Kakutani tower which we call a fat tower.

math.DS↗