Polynomial configurations in difference sets

DOI10.1016/J.JNT.2008.05.003zbMATH Open1228.11015arXiv0903.4504OpenAlexW2028158253MaRDI QIDQ999722

Neil Lyall, Ákos Magyar

Publication date: 10 February 2009

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

Abstract: We prove a quantitative version of the Polynomial Szemeredi Theorem for difference sets. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sarkozy (the simplest non-trivial case of the Polynomial Szemeredi Theorem asserting that the difference set of any subset of the integers of positive upper density necessarily contains a perfect square) and then applying a simple lifting argument.

Full work available at URL: https://arxiv.org/abs/0903.4504

zbMATH Keywords

difference setspolynomial configurationsSárközy's theorem

Mathematics Subject Classification ID

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Cites Work

• Title not available (Why is that?)
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• Difference sets without squares
• Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions
• Difference sets without \(\kappa\)-th powers
• On arithmetic structures in dense sets of integers
• Integer sets containing no arithmetic progressions
• The Hardy-Littlewood method.
• Intersective sets given by a polynomial
• Integer Sets Containing No Arithmetic Progressions
• On Sets of Natural Numbers Whose Difference Set Contains No Squares
• On difference sets of sequences of integers. III
• Difference sets and the primes
• Polynomial extensions of van der Waerden’s and Szemerédi’s theorems
• A new proof of Szemerédi's theorem

Related Items (11)

Multivariate Polynomial Values in Difference Sets ⋮ A maximal extension of the best-known bounds for the Furstenberg–Sárközy theorem ⋮ Binary Quadratic Forms in Difference Sets ⋮ Polynomial configurations in sets of positive upper density over local fields ⋮ Polynomial criterion for abelian difference sets ⋮ Sets of large dimension not containing polynomial configurations ⋮ Arithmetic structure in sparse difference sets ⋮ Bounds in a popular multidimensional nonlinear Roth theorem ⋮ A new proof of Sárközy's theorem ⋮ Intersective sets given by a polynomial ⋮ Sárközy's theorem for P-intersective polynomials

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