On the number of subsequence sums related to the support of a sequence in finite abelian groups

Abstract

Let G be a finite abelian group and S a sequence with elements of G. Let $\left|S\right|$ denote the length of S and $\mathrm{supp}\left(S\right)$ the set of all the distinct terms in S. For an integer k with $k\in [1,|S\left|\right]$, let ${\Sigma }_k\left(S\right)\subset G$ denote the set of group elements which can be expressed as a sum of a subsequence of S with length k. Let $\Sigma \left(S\right)={\cup }_{k=1}^{\left|S\right|}{\Sigma }_k\left(S\right)$ and ${\Sigma }_{\ge k}\left(S\right)={\cup }_{t=k}^{\left|S\right|}{\Sigma }_t\left(S\right)$. It is known that if $0\notin \Sigma \left(S\right)$, then $\left|\Sigma \right(S\left)\right|\ge \left|S\right|+\left|\mathrm{supp}\right(S\left)\right|-1$. In this paper, we determine the structure of a sequence S satisfying $0\notin \Sigma \left(S\right)$ and $\left|\Sigma \right(S\left)\right|=\left|S\right|+\left|\mathrm{supp}\right(S\left)\right|-1$. As a consequence, we can give a counterexample of a conjecture of Gao, Grynkiewicz, and Xia. Moreover, we prove that if $\left|S\right|>k$ and $0\notin {\Sigma }_{\ge k}\left(S\right)\cup \mathrm{supp}\left(S\right)$, then $\left|{\Sigma }_{\ge k}\right(S\left)\right|\ge \left|S\right|-k+\left|\mathrm{supp}\right(S\left)\right|$. Then we can give an alternative proof of a conjecture of Hamidoune, which was first proved by Gao, Grynkiewicz, and Xia.