Capacity dimension of the Brjuno set in ${\mathbb{C}}^n$

In this work, we prove that the complement of the Brjuno set in ${\mathbb{C}}^n$ has zero $C_{\sigma }$-capacity with respect to the kernel $k_{\sigma }(z,\xi )=\mathrm{\parallel }z-\xi {\mathrm{\parallel }}^{-2n+2}\mathrm{\mid }\log \mathrm{\parallel }z-\xi \mathrm{\parallel }{\mathrm{\mid }}^{\sigma }$ for any $\sigma >n$. In particular, it follows that it has zero $h_{\delta }$-Hausdorff measure with respect to the $h_{\delta }(t)=t^{2n-2}\mathrm{\mid }\log t{\mathrm{\mid }}^{-\delta }$, for any $\delta >n+1$. This generalizes a previous result of Sadullaev and the second author in dimension one to higher dimensions.