Infinite sumsets with many representations

Let $A$ be an infinite set of nonnegative integers. For $h\ge 2$, let $hA$ be the set of all sums of $h$ not necessarily distinct elements of $A$. If every sufficiently large integer in the sumset $hA$ has at least two representations, then $A(x)\ge (\log x)\mathrm{/}\log h)-w_0$, where $A(x)$ counts the number of integers $a\in A$ such that $1\le a\le x$.