On product of difference sets for sets of positive density

Abstract

In this paper we prove that given two sets $E_1,E_2 \subset \mathbb{Z}$ of positive density, there exists $k \geq 1$ which is bounded by a number depending only on the densities of $E_1$ and $E_2$ such that $k\mathbb{Z} \subset (E_1-E_1)\cdot(E_2-E_2)$. As a corollary of the main theorem we deduce that if $α,β> 0$ then there exist $N_0$ and $d_0$ which depend only on $α$ and $β$ such that for every $N \geq N_0$ and $E_1,E_2 \subset \mathbb{Z}_N$ with $|E_1| \geq αN, |E_2| \geq βN$ there exists $d \leq d_0$ a divisor of $N$ satisfying $d \, \mathbb{Z}_N \subset (E_1-E_1)\cdot(E_2-E_2)$.

6 pages; a new proof (due to I. Shkredov) of Lemma 2.1 that does not use Szemeredi's theorem has been added