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

Published 12 Oct 2018 in math.NT | (1810.05346v2)

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 $4^{{\wedge}\mathcal{A}=\mathbb{Z}_{n}$.}