Exact logarithmic Hausdorff and capacity exponents of the non-Brjuno set

Abstract

Let $\mathcal{N}$ be the set of non-Brjuno real numbers. We determine the exact critical exponent of $\mathcal{N}$ for both logarithmic Hausdorff measure and logarithmic capacity. For the gauges $h_δ(r)=(\log(1/r))^{-δ}$, the critical exponent is $2$: the $h_δ$-Hausdorff measure is infinite locally for $0<δ\leq2$ and vanishes globally for $δ>2$, while the logarithmic capacity is positive exactly for orders $0<s\leq2$. In particular, $\dim_{\mathrm{cap}}\mathcal{N}=2$.