Local characteristics of smoothing properties of endomorphisms of finite Abelian groups

Let $G$ be a finite Abelian group, $G^n$ be its $n$-fold Cartesian product, and $\vec\xi=(\xi_1,\xi_2,\dots,\xi_n)$ be a random element of $G^n$. We investigate the local characteristics of closeness of distribution of random element $H(\vec\xi\,)$, where $H\colon G^n\to G^m$, to the uniform distribution on $G^m$. Main results are connected with the case of independent identically distributed elements $\xi_1,\xi_2,\dots,\xi_n$ and endomorphism $H$ of group $G^n$ onto the group $G^m$.