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 $|S|$ denote the length of $S$ and $\mathrm{supp}(S)$ the set of all the distinct terms in $S$. For an integer $k$ with $k\in [1, |S|]$, let $\Sigma_{k}(S) \subset G$ denote the set of group elements which can be expressed as a sum of a subsequence of $S$ with length $k$. Let $\Sigma(S)=\cup_{k=1}^{|S|}\Sigma_{k}(S)$ and $\Sigma_{\geq k}(S)=\cup_{t=k}^{|S|}\Sigma_{t}(S)$. It is known that if $0\not\in \Sigma(S)$, then $|\Sigma(S)|\geq |S|+|\mathrm{supp}(S)|-1$. In this paper, we determine the structure of a sequence $S$ satisfying $0\notin \Sigma(S)$ and $|\Sigma(S)|= |S|+|\mathrm{supp}(S)|-1$. As a consequence, we can give a counterexample of a conjecture of Gao, Grynkiewicz, and Xia. Moreover, we prove that if $|S|>k$ and $0\not\in \Sigma_{\geq k}(S)\cup \mathrm{supp}(S)$, then $|\Sigma_{\geq k}(S)|\geq |S|-k+|\mathrm{supp}(S)|$. Then we can give an alternative proof of a conjecture of Hamidoune, which was first proved by Gao, Grynkiewicz, and Xia.