On the number of subsequences with a given sum in a finite abelian group

Suppose $G$ is a finite abelian group and $S$ is a sequence of elements in $G$. For any element $g$ of $G$, let $N_g(S)$ denote the number of subsequences of $S$ with sum $g$. The purpose of this paper is to investigate the lower bound for $N_g(S)$. In particular, we prove that either $N_g(S)=0$ or $N_g(S) \ge 2^{|S|-D(G)+1}$,where$D(G)$isthesmallestpositiveinteger$\ell$ such that every sequence over $G$ of length at least $\mathrm{\ell }$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.