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 $C_σ$-capacity with respect to the kernel $k_σ(z,ξ)=\|z-ξ\|^{-2n+2}|\log{\|z-ξ\||^σ}$ for any $σ>n$. In particular, it follows that it has zero $h_δ$-Hausdorff measure with respect to the $h_δ(t)=t^{2n-2}|\log{t}|^{-δ}$, for any $δ>n+1$. This generalizes a previous result of Sadullaev and the second author in dimension one to higher dimensions.