Work

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

Year: 2026

Type: preprint

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

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2608.29779

Authors: Bazarbaev, Sardor, Rakhimov, Karim

Institutions: Academy of Sciences Republic of Uzbekistan, National University of Uzbekistan

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7204943814

Cites: 3

Cited by:

Related to: 10