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Szemerédi's Theorem in the Primes

Part of: Sequences and sets

Published online by Cambridge University Press:  19 November 2018

Luka Rimanić  and
Julia Wolf
Luka Rimanić
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK (luka.rimanic@cantab.net; julia.wolf@cantab.net)
Julia Wolf
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK (luka.rimanic@cantab.net; julia.wolf@cantab.net)
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Abstract

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that, in fact, any subset of the primes of relative density tending to zero sufficiently slowly contains a three-term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of three-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemerédi theorem given by Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.

Keywords

Green–Tao theoremSzemerédi's theorempseudorandomnessarithmetic combinatorics

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 443 - 457
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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Footnotes

In memory of Kevin Henriot

References

1.Bloom, T., A quantitative improvement for Roth's theorem on arithmetic progressions, J. London Math. Soc. (2) 93(3) (2016), 643–663.Google Scholar
2.Conlon, D., Fox, J. and Zhao, Y., The Green–Tao theorem: an exposition, EMS Surv. Math. Sci. 1(2) (2014), 249–282.Google Scholar
3.Conlon, D., Fox, J. and Zhao, Y., A relative Szemerédi theorem, Geom. Funct. Anal. 25(3) (2015), 733–762.Google Scholar
4.Goldston, D. A., Pintz, J. and Yıldırım, C. Y., Primes in tuples. I, Ann. of Math. (2) 170(2) (2009), 819–862.Google Scholar
5.Gowers, W. T., A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11(3) (2001), 465–588.Google Scholar
6.Gowers, W. T., Decompositions, approximate structure, transference, and the Hahn–Banach theorem, Bull. London Math. Soc. 42(4) (2010), 573–606.Google Scholar
7.Green, B., Roth's theorem in the primes, Ann. of Math. (2) 161(3) (2005), 1609–1636.Google Scholar
8.Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167(2) (2008), 481–547.Google Scholar
9.Green, B. and Tao, T., New bounds for Szemerédi's theorem, III: A polylogarithmic bound for r 4 (n), preprint (arXiv:1705.01703, 2017).Google Scholar
10.Helfgott, H. A. and de Roton, A., Improving Roth's theorem in the primes, Int. Math. Res. Not. IMRN 2011(4) (2011), 767–783.Google Scholar
11.Henriot, K., On systems of complexity one in the primes, Proc. Edinb. Math. Soc. (2) 60(1) (2016), 133–163.Google Scholar
12.Naslund, E., On improving Roth's theorem in the primes, Mathematika 61(1) (2015), 49–62.Google Scholar
13.O'Bryant, K., Sets of integers that do not contain long arithmetic progressions, Electron. J. Combin. 18(1) Paper 59, 15, (2011).Google Scholar
14.Reingold, O., Trevisan, L., Tulsiani, M. and Vadhan, S., New proofs of the Green–Tao–Ziegler dense model theorem: an exposition, preprint (arXiv:0806.0381, 2008).Google Scholar
15.Tao, T. and Ziegler, T., The primes contain arbitrarily long polynomial progressions, Acta Math. 201(2) (2008), 213–305.Google Scholar
16.Varnavides, P., On certain sets of positive density, J. Lond. Math. Soc. 34 (1959), 358–360.Google Scholar
17.Zhao, Y., An arithmetic transference proof of a relative Szemerédi theorem, Math. Proc. Cambridge Philos. Soc. 156(2) (2014), 255–261.Google Scholar