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~$k \geq 2$, and~$A=\left{a_{0}, a_{1}, \ldots, a_{k}\right}$ is a finite set of integers such that~$0=a_{0}<a_{1}<\cdots<a_{k}$ and $\gcd\left(a_{1}, a_2,\ldots, a_{k}\right)=1,$ 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~$h \geq\sum_{i=2}^{k}(ta_{i}-1)-1.$ This improves a recent result of Nathanson with the bound $h \geq (k-1)\left(t a_{k}-1\right) a_{k}+1$.