Slow convergences of ergodic averages

TL;DR

This paper investigates the slow convergence rates of ergodic averages in ergodic automorphisms, building on Krengel's work, and provides new results answering a question posed by Podvigin.

Contribution

It offers new proofs and results on the slow convergence of ergodic averages, extending Krengel's findings and addressing Podvigin's question.

Findings

Demonstrates existence of indicator functions with arbitrarily slow convergence rates

Provides new proofs of slow convergence results

Answers a specific open question in ergodic theory

Abstract

Birkhoff's theorem states that for an ergodic automorphism, the time averages converge to the space average. Given sequence $\psi(n)\to+0$, U. Krengel proved that for any ergodic automorphism there is an indicator such that the corresponding time averages converged a.e. with a rate slower than $\psi$. We prove again similar statements answering a question of I. Podvigin in passing.