Improved bounds for arithmetic progressions in product sets

2015/02/12 by Dmitry Zhelezov, Zhelezov, Dmitry
Mathematics · #11B25 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1502.03704

openalex publication_date 2015/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B be a set of natural numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = \bb'| b, b' ∈ B\ cannot be greater than O(n log n) which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers we improve the bound to O_ε(n^{1 + ε}) for arbitrary ε> 0 assuming the GRH.

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