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

TL;DR

This paper establishes that the complements of the Brjuno and Perez-Marco sets have zero capacity with respect to specific logarithmic kernels, advancing understanding of their geometric and potential-theoretic properties.

Contribution

It proves zero capacity results for the complements of Brjuno and Perez-Marco sets using novel logarithmic kernels, revealing new potential-theoretic characteristics.

Findings

Complement of Brjuno set has zero capacity w.r.t. a specific kernel.

Complement of Perez-Marco set has zero capacity w.r.t. a different kernel.

Results deepen understanding of the geometric structure of these sets.

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 $\sigma>2$.