A relative Szemerédi theorem
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39_2015_Article_324.pdf
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Author(s)
Conlon, David
Fox, Jacob
Zhao, Yufei
Date Issued
March 2015
Journal
Geometric and Functional Analysis
Publisher
Springer Basel
Citation
Conlon, David, Jacob Fox, and Yufei Zhao. “A Relative Szemerédi Theorem.” Geometric and Functional Analysis 25, no. 3 (March 17, 2015): 733–762.
Version
Author's final manuscript
Abstract
The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for k-term arithmetic progressions in pseudorandom subsets of Z[subscript N] of density N[superscript −c[subscript k]]. The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemerédi theorem.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00039-015-0324-9
