On the rate of convergence of $\mathbb{R}^d$-ergodic averages constructed over a strictly convex set

TL;DR

This paper establishes a spectral criterion for the convergence rates of ergodic averages over strictly convex sets in , showing that these rates are independent of the particular convex set used.

Contribution

It introduces a spectral criterion for homogeneous convergence rates of -ergodic means over strictly convex sets, extending Hertz's Fourier transform asymptotics.

Findings

Convergence rates are independent of the specific convex set.

A spectral criterion for homogeneous convergence is derived.

The results connect Fourier transform asymptotics with ergodic averages.

Abstract

For $\mathbb{R}^d$-ergodic means constructed over a strictly convex set, a spectral criterion for homogeneous rates of convergence is obtained. From Hertz's result on the asymptotics of the Fourier transform of the indicator of a strictly convex set, it follows that the rate of convergence does not depend on the specific form of such a set.