On subset sums of ${\mathbb{Z}}_n^{\times }$ which are equally distributed modulo $n$

In this note, we provide some results concerning the structure of a set $A\subseteq {\mathbb{Z}}_n^{\times }$, which has non-empty subset sums equally distributed modulo $n$. Here, ${\mathbb{Z}}_n^{\times }$ denotes the set which contains all the invertible elements of the ring ${\mathbb{Z}}_n$. In particular, we prove that if $n=q$ is a power of an odd prime, then $A$ is a union of sets of the form $\{a\cdot (\pm 2^i)\}$. Additionally, we count the number of subsets of ${\mathbb{Z}}_q^{\times }$ with non-empty subset sums equally distributed modulo $q$.