ON PRODUCT OF DIFFERENCE SETS FOR SETS OF POSITIVE DENSITY

In this paper we prove that given two sets E-1, E-2 subset of Z of positive density, there exists k >= 1 which is bounded by a number depending only on the densities of E-1 and E-2 such that kZ subset of (E-1 - E-1) subset of ( 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 >= N-0 and E-1, E-2 subset of Z(N) with vertical bar E-1 vertical bar >= alpha N, vertical bar E-2 vertical bar >= beta N there exists d <= d(0) a divisor of N satisfying dZ(N) subset of (E-1 - E-1) center dot (E-2 - E-2).