Ergodic averages with deterministic weights

TL;DR

This paper investigates the convergence properties of ergodic averages with deterministic weights, focusing on sequences with specific bounded oscillation characterized by a parameter , and establishes conditions under which these averages converge.

Contribution

It introduces a new framework for analyzing the convergence of weighted ergodic averages with deterministic weights under specific boundedness conditions.

Findings

Established bounds for convergence based on the parameter .

Defined the infimum (, u) for the class of sequences satisfying the boundedness condition.

Provided conditions under which ergodic averages with deterministic weights converge.

Abstract

The purpose of this paper is to study ergodic averages with deterministic weights. More precisely we study the convergence of the ergodic averages of the type $\frac{1}{N} \sum_{k=0}^{N-1} \theta (k) f \circ T^{u_k}$ where $\theta = (\theta (k) ; k\in \NN)$ is a bounded sequence and $u = (u_k ; k\in \NN)$ a strictly increasing sequence of integers such that for some $\delta<1$ $$ S_N (\theta, u) := \sup_{\alpha \in \pRR} | \sum_{k=0}^{N-1} \theta (k) \exp (2i\pi \alpha u_k) | = O (N^{\delta}) \leqno{({\cal H}_1)} $$ i.e., there exists a constant $C$ such that $S_N (\theta, u) \leq C N^{\delta} $. We define $\delta (\theta, u)$ to be the infimum of the $\delta $ satisfying $\H_1$ for $\theta $ and $u$.