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arXiv · 2507.15980

On the capacity dimensions of the Brjuno and Perez-Marco sets

Nurali Akramov·Abduvahhob Ashirov
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Abstract

In this work, we prove that the complement of the Brjuno set $\mathcal{B}$ has a zero capacity with respect to the kernel $k^1_\sigma(z,\xi)=\ln^2{|z-\xi|}\left|\ln{\ln{\left(e+\frac{1}{|z-\xi|}\right)}}\right|^\sigma$ for any $\sigma > 2$. Similarly, the complement of the Perez-Marco set $\mathcal{PM}$ has a zero capacity with respect to the kernel $k^2_\sigma(z,\xi) = \ln^{2}{\ln\left(e+\frac{1} {\left| {z - \xi }\right|}\right)} \cdot\ln^{\sigma}{\ln\ln\left(e^3+\frac{1} {\left| {z - \xi }\right|}\right)}$ for any $\sigma>2$.

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Nurali Akramov, Abduvahhob Ashirov. 2025-07-21. On the capacity dimensions of the Brjuno and Perez-Marco sets. https://arxiv.org/abs/2507.15980

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