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Exact logarithmic Hausdorff and capacity exponents of the non-Brjuno set
Abstract
Let 𝒩 be the set of non-Brjuno real numbers. We determine the exact critical exponent of 𝒩 for both logarithmic Hausdorff measure and logarithmic capacity. For the gauges hδ(r)=(log(1/r))−δ, the critical exponent is $2$: the hδ-Hausdorff measure is infinite locally for 0<δ≤2 and vanishes globally for δ>2, while the logarithmic capacity is positive exactly for orders 0<s≤2. In particular, dimcap𝒩=2.
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BibTeX
@misc{bazarbaev2026exact,
title = {{Exact logarithmic Hausdorff and capacity exponents of the non-Brjuno set}},
author = {Sardor Bazarbaev and Karim Rakhimov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29779},
eprint = {2608.29779},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29779},
}
RIS
TY - UNPB
TI - Exact logarithmic Hausdorff and capacity exponents of the non-Brjuno set
AU - Sardor Bazarbaev
AU - Karim Rakhimov
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29779
UR - https://arxiv.org/abs/2608.29779
AB - Let 𝒩 be the set of non-Brjuno real numbers. We determine the exact critical exponent of 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≤2 and vanishes globally for δ>2, while the logarithmic capacity is positive exactly for orders 0<s≤2. In particular, dim_cap𝒩=2.
ER -