<i>h</i>-Fold Sums from a Set with Few Products

Abstract

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 $h(A.A)=A.A+\dots+A.A$.