Infinite sumsets with many representations

Let $A$ be an infinite set of nonnegative integers. For $h \geq 2$, 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 $A(x) \geq (\log x)/\log h)-w_0$, where $A(x)$ counts the number of integers $a \in A$ such that $1 \leq a \leq x$.