It is well known that the Gauss map $T: [0,1] \setminus \mathbb{Q} \to [0,1] \setminus \mathbb{Q}$ given by $$T(x)= \frac{1}{x} \mod 1$$ has an absolutely continuous invariant probability measure $\muT$ given by $$\muT(A)= \frac{1}{\log 2} \int_A \frac{1}{1+x} dx.$$
Let $\mu{\mathbf{p}}$ denote the Bernoulli measure associated to the countable probability vector $\mathbf{p}$, projected to $[0,1]$ in the usual way. Kifer, Peres and Weiss showed that the Bernoulli measures for the Gauss map satisfy a \emph{dimension gap} meaning that there exists $c>0$ such that \begin{eqnarray} (1)\quad\sup{\mathbf{p}} \dim{\mathrm{H}} \mu{\mathbf{p}} < 1- c. \label{dimgap} \end{eqnarray} Moreover, they showed that $c \geq 10^{-7}$. Their proof was based on considering sets of large deviations for the asymptotic frequency of certain digits from the one prescribed by $\mu_T$.
In this talk we will discuss an alternative proof of (1) which instead reduces to obtaining good lower bounds on the asymptotic variance of a class of potentials.
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