A new proof of the dimension gap for the Gauss map

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.