Some inverse results of sumsets

Abstract

Let $h\ge 2$ and $A=\{a_0,a_1,\dots ,a_{k-1}\}$ be a finite set of integers. It is well-known that $\left|hA\right|=hk-h+1$ if and only if $A$ is a $k$-term arithmetic progression. In this paper, we give some nontrivial inverse results of the sets $A$ with some extrema the cardinalities of $hA$.