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      Discrete spheres and arithmetic progressions in product sets

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          Abstract

          We prove that if \(B\) is a set of \(N\) positive integers such that \(B\cdot B\) contains an arithmetic progression of length \(M\) then \(N\geq \pi(M) + M^{2/3-o(1)}\). On the other hand, there are examples for which \(N< \pi(M)+ M^{2/3}\). This improves previously known bounds of the form \(N = \Omega(\pi(M))\) and \(N = O(\pi(M))\), 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 \(\mathbb{F}_3^n\) 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 \(|A| \gg n^2\).

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          Journal
          2015-10-19
          Article
          1510.05411
          e91c7488-e04b-45ae-84e2-ccc45b99e7f0

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          11B25
          math.NT

          Number theory
          Number theory

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