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

Abstract

Let $h\ge 2$ 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 $|\mathcal{A}|\ge 0.4045n$ 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 $4^{\wedge }\mathcal{A}={\mathbb{Z}}_n$.