  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.
