On the structure of the $h$-fold sumsets

Let~$A$ be a set of nonnegative integers. Let~$(hA{)}^{(t)}$ 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~$A=\{a_0,a_1,\dots ,a_k\}$ is a finite set of integers such that~$0=a_0<a_1<\cdots <a_k$ and $\gcd\left(a_{1}, a_2,\ldots, a_{k}\right)=1,$thenthereexistintegers$c_{t},d_{t}$ and sets~$C_t\subseteq [0,c_t-2]$, $D_t\subseteq [0,d_t-2]$ such that $$(hA{)}^{(t)}=C_t\cup [c_t,h{a}_k-d_t]\cup (h{a}_{k-1}-D_t)$$ for all~$h\ge {\sum }_{i=2}^k(t{a}_i-1)-1.$ This improves a recent result of Nathanson with the bound $h \geq (k-1)\left(t a_{k}-1\right) a_{k}+1$.