An L^1 Ergodic Theorem for Sparse Random Subsequences

TL;DR

This paper establishes an L^1 ergodic theorem for sparse random subsequences, demonstrating that such sequences can be nearly as sparse as perfect squares while still maintaining ergodic properties.

Contribution

It introduces a new L^1 subsequence ergodic theorem for randomly selected sequences, extending ergodic theory to sparser sequences than previously known.

Findings

Proves an L^1 ergodic theorem for random subsequences

Shows sparse sequences can be universally L^1-good

Establishes a deterministic condition implying a weak maximal inequality

Abstract

We prove an L^1 subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L^1-good sequences nearly as sparse as the set of squares. In the process, we prove that a certain deterministic condition implies a weak maximal inequality for a sequence of \ell^1 convolution operators.