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 $h_{\delta }(r)=(\log (1/r){)}^{-\delta }$, the critical exponent is $2$: the $h_{\delta }$-Hausdorff measure is infinite locally for $0<\delta \le 2$ and vanishes globally for $\delta >2$, while the logarithmic capacity is positive exactly for orders $0<s\le 2$. In particular, ${\dim }_{\mathrm{cap}}\mathcal{N}=2$.