On n-sum of an abelian group of order n

TL;DR

This paper proves a conjecture regarding the sumsets of sequences in finite abelian groups, establishing a lower bound on the size of certain sumsets or the inclusion of zero, thus advancing additive combinatorics.

Contribution

It confirms Hamidoune's 2000 conjecture on the structure of sumsets in finite abelian groups, providing a new lower bound for the sumset size or zero inclusion.

Findings

Either zero is in the sumset or the sumset size is at least k + t - 1.

The result applies to sequences with n + k elements containing t distinct elements.

The proof confirms a longstanding conjecture in additive number theory.

Abstract

Let $G$ be an additive finite abelian group of order $n$, and let $S$ be a sequence of $n+k$ elements in $G$, where $k\geq 1$. Suppose that $S$ contains $t$ distinct elements. Let $\sum_n(S)$ denote the set that consists of all elements in $G$ which can be expressed as the sum over a subsequence of length $n$. In this paper we prove that, either $0\in \sum_n(S)$ or $|\sum_n(S)|\geq k+t-1.$ This confirms a conjecture by Y.O. Hamidoune in 2000.