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arXiv · 2608.29779

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

Sardor Bazarbaev·Karim Rakhimov
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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 $0<\delta\leq2$ and vanishes globally for $\delta>2$, while the logarithmic capacity is positive exactly for orders $0<s\leq2$. In particular, $\dim_{\mathrm{cap}}\mathcal{N}=2$.

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Sardor Bazarbaev, Karim Rakhimov. 2026-08-30. Exact logarithmic Hausdorff and capacity exponents of the non-Brjuno set. https://arxiv.org/abs/2608.29779

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