On the structure of the $h$-fold sumsets

Abstract

Let $A$ be a set of nonnegative integers. Let $(h A)^{(t)}$ be the set of all integers in the sumset $hA$ that have at least $t$ representations as a sum of $h$ elements of $A$. In this paper, we prove that, if $k \ge 2$, and $A=\left\lbrace a_{0}, a_{1}, \dots , a_{k}\right\rbrace $ is a finite set of integers such that $0=a_{0} and $\gcd \left(a_{1}, a_2,\dots , a_{k}\right)=1,$ then there exist integers $c_{t},d_{t}$ and sets $C_{t}\subseteq [0, c_{t}-2]$, $D_{t} \subseteq [0, d_{t}-2]$ such that \[ (h A)^{(t)}=C_{t} \cup \left[c_{t}, h a_{k}-d_{t}\right] \cup \left(h a_{k-1}-D_{t}\right) \] for all $h \ge \sum _{i=2}^{k}(ta_{i}-1)-1.$ This improves a recent result of Nathanson with the bound $h \ge (k-1)(t a_{k}-1) a_{k}+1$.