Capacity of the α-Brjuno-Rüssmann set

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