The sum-product problem for integers with few prime factors

Abstract

It was asked by E. Szemerédi if, for a finite set $A\subset \mathbb{Z}$, one can improve estimates for $\max \{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega (a)\le k$. In this paper, answer Szemerédi's question in the affirmative by showing that this maximum is of order $|A{|}^{\frac{5}{3}-o(1)}$ provided $k\le (\log |A|{)}^{1-\epsilon }$ for some $\epsilon >0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A{|}^{o(1)}$.

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