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 $k\ge 1$ 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 $\alpha ,\beta >0$ then there exist $N_0$ and $d_0$ which depend only on $\alpha$ and $\beta$ such that for every $N\ge N_0$ and $E_1,E_2\subset {\mathbb{Z}}_N$ with $|E_1|\ge \alpha N,|E_2|\ge \beta N$ there exists $d\le d_0$ 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

References in corpus (1)

• New results on sum-products in R