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Discrete spheres and arithmetic progressions in product sets

Dmitrii Zhelezov (Institutionen för matematiska vetenskaper)
Acta Arithmetica (0065-1036). Vol. 178 (2017), 3, p. 235-248.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: Arithmetic progressions; Multiplicative basis ; Product sets

Denna post skapades 2015-12-03. Senast ändrad 2017-07-12.
CPL Pubid: 227090


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