The restricted sumsets in $\mathbb{Z}_n$

TL;DR

This paper investigates the structure of restricted sumsets in cyclic groups, establishing conditions under which the sumset of 4 or 5 distinct elements covers the entire group, with implications for additive combinatorics.

Contribution

It provides new bounds and conditions ensuring that restricted sumsets of size 4 and 5 cover the entire cyclic group, extending previous results in additive number theory.

Findings

If ||  0.4045n and n is odd, then 4^{}=n;

For even n and  close to n/4, 4^{}=n;

New bounds for sumset coverage in cyclic groups.

Abstract

Let $h\geq 2$ be a positive integer. For any subset $\mathcal{A}\subset \mathbb{Z}_n$, let $h^{\wedge}\mathcal{A}$ be the set of the elements of $\mathbb{Z}_n$ which are sums of $h$ distinct elements of $\mathcal{A}$. In this paper, we obtain some new results on $4^{\wedge}\mathcal{A}$ and $5^{\wedge}\mathcal{A}$. For example, we show that if $|\mathcal{A}|\geq 0.4045n$ and $n$ is odd, then $4^{\wedge}\mathcal{A}=\mathbb{Z}_{n}$; Under some conditions, if $n$ is even and $|\mathcal{A}|$ is close to $n/4$, then $4^{\wedge}\mathcal{A}=\mathbb{Z}_{n}$.