Discrete spheres and arithmetic progressions in product sets
Journal article, 2017

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 𝔽n3 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.

Arithmetic progressions

Product sets

Multiplicative basis


Dmitrii Zhelezov

Chalmers, Mathematical Sciences

University of Gothenburg

Acta Arithmetica

0065-1036 (ISSN)

Vol. 178 3 235-248

Subject Categories


Discrete Mathematics


Basic sciences



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