On the number of weighted subsequences with zero-sum in a finite abelian group

Abstract

Suppose $G$ is a finite abelian group and $S=g_1\cdots g_l$ is a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq \mathbb{Z}\backslash \left\{0\right\}$, let $N_{A,g}(S)$ denote the number of subsequences $T={\prod }_{i\in I}{g}_i$ of $S$ such that ${\sum }_{i\in I}{a}_i{g}_i=g$ , where $I\subseteq \left\{1,\dots ,l\right\}$ and $a_i\in A$. The purpose of this paper is to investigate the lower bound for $N_{A,0}(S)$. In particular, we prove that $N_{A,0}(S)\ge 2^{|S|-D_A(G)+1}$, where $D_A(G)$ is the smallest positive integer $l$ such that every sequence over $G$ of length at least $l$ has a nonempty $A$-zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.

arXiv commentThis paper was replaced by paper number 1811.03890 "On the number of fully weighted zero-sum subsequences"