# BEHAVIOR OF ERGODIC AVERAGES ALONG A SUBSEQUENCE AND THE GRID METHOD

2023

Dissertation

## Degree Name

Doctor of Philosophy

## Department

Mathematical Sciences

Máté Wierdl

Béla Bollobás

## Committee Member

James T. Campbell

## Committee Member

Randall McCutcheon

Máté Wierdl

## Abstract

\hspace{.2cm} In the first part of our thesis, we introduce the `grid method' to prove that a very strong type of oscillation occurs for the averages obtained by sampling a flow along the sequence of times of the form $\{n^\alpha: n\in \setN\}$, where $\alpha$ is a {positive non-integer rational number}. More precisely, we show that in every aperiodic dynamical system $(X, \Sigma, \mu, T^t)$, we can find a set $E$ of arbitrarily small measure so that $\limsup_N\frac1N \sum_{n\le N}\setone_E(T^{n^\alpha}x)=1$ and $\liminf_N\frac1N \sum_{n\le N}\setone_E(T^{n^\alpha}x)=0$ almost everywhere. Such behavior of a sequence is known as the \emph{strong sweeping out} property. Using the grid method, we will give an example of a general class of sequences which satisfy the {strong sweeping out} property.\\ \hspace{.2cm} In the second part of our thesis, we deal with ergodic averages along a \emph{sublacunary sequence} of positive integers. An increasing sequence $(a_n)$ of positive integers which satisfies $\frac{a_{n+1}}{a_n}>1+\eta$ for some positive $\eta$ is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence is strong sweeping out. As a further application of the grid method, we improve this result by showing that if $(a_n)$ satisfies only $\frac{a_{n+1}}{a_n}>1+\frac1{(\log\log n)^{1-\eta}}$ for some positive $\eta$ then it is already strong sweeping out.\\ \hspace{.2cm} In the third part, we will prove that if an integer sequence $(b_n)$ grows slower than any positive power of $n$, and $(a_n)$ is a small perturbation of $(b_n)$, then $(a_n)$ is strong sweeping out. This result is sharp. That means for any $c>0$, the ergodic averages along $\lfloor n^c \rfloor$ converge almost everywhere for all $f\in L^p$ when $p>1$.

Data is provided by the student.

## Library Comment

Dissertation or thesis originally submitted to ProQuest

Open Access

COinS