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

TL;DR

This paper establishes a lower bound on the number of subsequences with a given sum in finite abelian groups, linking it to zero-sum subsequence properties and characterizing extremal cases.

Contribution

It introduces a new lower bound for the count of subsequences with a specific sum in finite abelian groups and characterizes extremal sequences where equality is achieved.

Findings

Lower bound: N_g(S) ≥ 2^{|S|-D(G)+1} when N_g(S) > 0

Characterization of extremal sequences achieving equality

Connection between subsequence sums and zero-sum subsequence properties

Abstract

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 $D(G)$ is the smallest positive integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a nonempty zero-sum subsequence. We also characterize the structures of the extremal sequences for which the equality holds for some groups.