A $U$-set for system of character of the zero-dimensional group under convergent over cubes

In this work we consider system of characters of the compact zero-dimensional group $G$ and study uniqueness sets for $N$-fold multiple series for system of character a zero-dimensional group under convergent over cubes (in other words, $U$-sets). We proof that every finite set is a $U$-set and show that countable set with only one limit point is a $U$-set.