Pointwise convergence of double ergodic averages along certain non-polynomial sequences

TL;DR

The paper proves that for a measure-preserving system $(X,\mathcal{X},\mu,T)$ and $f,g\in L^{\infty}(\mu)$, the double ergodic averages along the non-polynomial sequences $\lfloor \alpha n^{c}\rfloor$ and $\lfloor \beta n^{c}\rfloor$ converge pointwise almost everywhere when $c\in(1,23/22)$ and $|\alpha|\neq|\beta|$. The authors develop a flow-based approximation framework, adapting the Daskalakis method, and treat the rational and irrational ratios $\alpha/\beta$ separately, aided by Diophantine approximation. They identify the characteristic factor $\mathcal{Z}_{2}(T)$: if $f$ or $g$ is orthogonal to $\mathcal{Z}_{2}(T)$, the limit is zero, and the limit in general depends only on $\alpha/\beta$. In addition, a multidimensional version with commuting transformations is established, along with a corresponding $L^{p}$-maximal inequality (Theorem TC). Overall, the work extends Bourgain-type pointwise convergence results to non-polynomial time sequences and provides a robust framework for understanding the limiting behavior in terms of low-complexity factors, with potential implications for irregular ergodic averages in higher dimensions.

Abstract

Fix $c\in (1,23/22)$. Let $α$ and $β$ be two distinct non-zero real numbers with $|α|\neq |β|$. It is shown that for any measure preserving system $(X,\mathcal{X},μ,T)$ and any $f,g\in L^{\infty}(μ)$, the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{\lfloor αn^c \rfloor}x)g(T^{\lfloor βn^c \rfloor}x) \end{equation*} exists for $μ$-a.e. $x\in X$. Meanwhile, a multidimensional version of the above result is also presented.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4