A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times
Ben Krause
TL;DR
The paper addresses pointwise convergence of bilinear ergodic averages along integer parts of Kronecker sequences, proving a unified result that covers both Bourgain's Double Recurrence and Return Times theorems. It develops a hard-analytic, time-frequency framework that transfers the problem to the integers, quantifies convergence via oscillation and maximal bounds, and organizes Fourier multipliers through forest and branch structures. Central contributions include a transference-to-integers reduction, a robust entropic and multi-scale decomposition, and a complete maximal-oscillation estimate that yields almost everywhere convergence with identical convergence sets for all rational weights. The methodological synthesis—combining Wiener-Wintner type reductions, entropy arguments, dyadic multiplier theory, and a detailed floor-function treatment—has broad implications for multi-linear ergodic theory and harmonic analysis on discrete and continuous settings, enabling new reach in pointwise convergence problems for complex or non-smooth dynamical or spectral data.
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 $α\in \mathbb{R}$: First, for each $σ$-finite measure-preserving system, $(X,μ,T)$, and each $f,g \in L^{\infty}(X)$, for each $γ\in \mathbb{Q}$ the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor αn \rfloor } f \cdot T^{\lfloor γn \rfloor} g \] converge $μ$-a.e.; Second, for each aperiodic and countably generated measure-preserving system, $(Y,ν,S)$, and each $g \in L^{\infty}(Y)$, there exists a subset $Y_{g} \subset Y$ with $ν(Y_{g})= 1$ so that for all $γ\in \mathbb{Q}$ and $ω\in Y_{g}$, for any auxiliary $σ$-finite measure-preserving system $(X,μ,T)$, 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 $μ$-a.e. Moreover, in both cases the sets of convergence are identical for all $γ\in \mathbb{Q}$.