A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times

TL;DR

This paper develops a unified method to extend key ergodic theorems involving double recurrence and return times to systems with integer parts of Kronecker sequences, emphasizing stopping times and metric entropy.

Contribution

It introduces a unified approach to prove convergence of bilinear ergodic averages involving integer parts of sequences, extending Bourgain's theorems to new settings.

Findings

Almost everywhere convergence of bilinear averages for integer-part sequences.

Extension of Bourgain's theorems to Kronecker sequences.

Convergence sets are identical for all rational parameters.

Abstract

We present a unified approach to extensions of Bourgain's Double Recurrence Theorem and Bourgain's Return Times Theorem to integer parts of the Kronecker sequence, emphasizing stopping times and metric entropy. Specifically, we prove the following two results for each $\alpha \in \mathbb{R}$: First, for each $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and each $f,g \in L^{\infty}(X)$, for each $\gamma \in \mathbb{Q}$ the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor \alpha n \rfloor } f \cdot T^{\lfloor \gamma n \rfloor} g \] converge $\mu$-a.e.; Second, for each aperiodic and countably generated measure-preserving system, $(Y,\nu,S)$, and each $g \in L^{\infty}(Y)$, there exists a subset $Y_{g} \subset Y$ with $\nu(Y_{g})= 1$ so that for all $\gamma \in \mathbb{Q}$ and $\omega \in Y_{g}$, for any auxiliary $\sigma$-finite measure-preserving system $(X,\mu,T)$, and any $f \in L^{\infty}(X)$, the ``return-times" averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor \alpha n \rfloor} f \cdot S^{\lfloor \gamma n \rfloor } g(\omega) \] converge $\mu$-a.e. Moreover, in both cases the sets of convergence are identical for all $\gamma \in \mathbb{Q}$.