h-Fold Sums from a Set with Few Products

In the present paper we show that if $A$ is a set of $n$ real numbers and the product set $A.A$ has at most $n^{1+\varepsilon}$ elements, then the $h$-fold sumset $hA$ has at least $n^{\log(h/2)/2\log2+1/2-f_h(\varepsilon)}$ elements, where $f_h(c)\to0$ as $c\to0$. We also prove results on the $h$-fold sumset $h(A.A)=A.A+\dots+A.A$.

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