Arithmetic structure in sparse difference sets

DOI10.1016/J.JNT.2010.02.016zbMATH Open1210.11021arXiv1004.2723OpenAlexW2963665949MaRDI QIDQ984846

Neil Lyall, Mariah Hamel, Katherine Thompson, Nathan Walters

Publication date: 20 July 2010

Published in: Journal of Number Theory (Search for Journal in Brave)

Abstract: Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration $v_1,...,v_{e}ll$ of vectors in $mathbb{Z}^d$, we show that if $Asubset[1,N{]}^d$ with $|A|/N^{d}geqC{N}^{-1/ell}$, then there necessarily exists $re0$ such that $r{v}_1,...,r{v}_{e}llsubseteqA-A$.

Full work available at URL: https://arxiv.org/abs/1004.2723

zbMATH Keywords

difference setsarithmetic progressions

Mathematics Subject Classification ID

Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) (05B10) Arithmetic progressions (11B25)

Cites Work

• Unnamed Item
• Difference sets without squares
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• Arithmetic progressions in sumsets
• Arithmetic progressions in sumsets

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ARITHMETIC PROGRESSIONS IN SUMSETS AND DIFFERENCE SETS ⋮ On theorems of Wirsing and Sanders

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