Weighted multiple ergodic averages via analytic observables over $ \mathbb{T}^\infty $: Is exponential pointwise convergence universal?

TL;DR

This paper proves that exponential pointwise convergence of weighted multiple ergodic averages over infinite-dimensional tori is universal under certain conditions, using an innovative accelerated weighting method and addressing key open questions.

Contribution

It introduces a novel accelerated weighting approach that ensures exponential convergence in ergodic averages over $\mathbb{T}^\infty$, resolving longstanding open problems in the field.

Findings

Established polynomial and exponential pointwise convergence under general conditions.

Demonstrated the universality of exponential convergence excluding nearly rational rotations.

Constructed counterexamples emphasizing the importance of nonresonance conditions.

Abstract

By employing an accelerated weighting method, we establish arbitrary polynomial and exponential pointwise convergence for multiple ergodic averages under general balancing conditions in both discrete and continuous settings, including quasi-periodic and almost periodic cases. We also present joint Diophantine rotations as explicit applications. Specifically, for the first time, by excluding nearly rational rotations with zero measure, we address the fundamental question of whether exponential pointwise convergence via analytic observables is universal, even when multiplicatively averaging over the infinite-dimensional torus $ \mathbb{T}^\infty $. We achieve this by introducing an innovative approach that effectively overcomes the previous difficulties. Moreover, by constructing counterexamples concerning multiple ergodicity, we highlight the indispensability of the joint nonresonance and establish the optimality of our weighting method in preserving rapid convergence. We also provide numerical simulations and analysis to further illustrate and validate our results.