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 $\{ a\cdot(\pm2^i)\}$. Additionally, we count the number of subsets of $\mathbb{Z}_q^{\times}$ with non-empty subset sums equally distributed modulo $q$.