Discrete spheres and arithmetic progressions in product sets

Abstract

We prove that if B is a set of N positive integers such that B⋅B contains an arithmetic progression of length M then N≥π(M)+M2/3−o(1). On the other hand, there are examples for which N<π(M)+M2/3. This improves previously known bounds of the form N=Ω(π(M)) and N=O(π(M)), respectively. The main new tool is a reduction of the original problem to the question of an approximate additive decomposition of the 3-sphere in &#120125;n3 which is the set of 0-1 vectors with exactly three non-zero coordinates. Namely, we prove that such a set cannot be contained in a sumset A+A unless |A|≫n2.