On product of difference sets for sets of positive density

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In this paper we prove that given two sets E₁,E₂ ⊂ Z of positive density, there exists k ≥ 1 which is bounded by a number depending only on the densities of E₁ and E₂ such that kZ ⊂ (E₁-E₁)·(E₂-E₂). As a corollary of the main theorem we deduce that if α,β> 0 then there exist N₀ and d₀ which depend only on α and β such that for every N ≥ N₀ and E₁,E₂ ⊂ Zₙ with |E₁| ≥ αN, |E₂| ≥ βN there exists d ≤ d₀ a divisor of N satisfying d Zₙ ⊂ (E₁-E₁)·(E₂-E₂).