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

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 $\hat{I}\pm \in \mathbb{R}$: First, for each $\ddot{I}ƒ$-finite measure-preserving system, $(X,\hat{I}¼,T)$, and each $f,g\in L^{\infty }(X)$, for each ${\hat{I}}^3\in \mathbb{Q}$ the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor αn \rfloor } f \cdot T^{\lfloor γn \rfloor} g \] converge $\hat{I}¼$-a.e.; Second, for each aperiodic and countably generated measure-preserving system, $(Y,\hat{I}½,S)$, and each $g\in L^{\infty }(Y)$, there exists a subset $Y_g\subset Y$ with $\hat{I}½(Y_g)=1$ so that for all ${\hat{I}}^3\in \mathbb{Q}$ and $\ddot{I}‰\in Y_g$, for any auxiliary $\ddot{I}ƒ$-finite measure-preserving system $(X,\hat{I}¼,T)$, and any $f\in L^{\infty }(X)$, the ``return-times" averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor αn \rfloor} f \cdot S^{\lfloor γn \rfloor } g(ω) \] converge $\hat{I}¼$-a.e. Moreover, in both cases the sets of convergence are identical for all ${\hat{I}}^3\in \mathbb{Q}$.