A new proof of the dimension gap for the Gauss map

TL;DR

This paper presents a new proof of the dimension gap for Bernoulli measures of the Gauss map, utilizing thermodynamic formalism to establish uniform bounds on asymptotic variance.

Contribution

It introduces a novel proof technique for the dimension gap, connecting thermodynamic formalism with variance bounds in the context of the Gauss map.

Findings

Establishes a uniform lower bound on asymptotic variance for certain potentials.

Provides a new proof of the dimension gap without relying on previous methods.

Connects thermodynamic formalism to measure dimension properties.

Abstract

Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map $T(x)= \frac{1}{x} \mod 1$ satisfy a `dimension gap' meaning that for some $c>0$, $\sup_{\mathbf{p}} \dim \mu_{\mathbf{p}} <1-c$, where $\mu_{\mathbf{p}}$ denotes the (pushforward) Bernoulli measure for the countable probability vector $\mathbf{p}$. In this paper we propose a new proof of the dimension gap. By using tools from thermodynamic formalism we show that the problem reduces to obtaining uniform lower bounds on the asymptotic variance of a class of potentials.