The Skitovich-Darmois theorem for discrete and compact totally disconnected Abelian groups

Let $X$ be an Abelian group of the form $X=\mathbb{R}^m\times K\times D$, where $m\geq 0$, $K$ is a compact totally disconnected group of the special form, $D$ is a discrete group. Let $\xi_i, i=1,2,...,n,n\geq 2,$ be independent random variables with values in $X$ and distributions $\mu_i$, and $\alpha_{ij},i,j=1,2,...,n,$ be topological automorphisms of $X$. We prove that the independence of the linear forms $L_j=\sum_{i=1}^{n}\alpha_{ij}\xi_i,j=1,2,...,n,$ implies that all $\mu_i$ are convolutions of Gaussian and idempotent distributions. This theorem can be considered as a generalization for the group X of the well-known Skitovich-Darmois theorem for $n$ linear forms.