Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-l69ms Total loading time: 0.205 Render date: 2022-08-19T08:27:45.063Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Szemerédi's Theorem in the Primes

Published online by Cambridge University Press:  19 November 2018

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)

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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), 643663.CrossRefGoogle Scholar
2.Conlon, D., Fox, J. and Zhao, Y., The Green–Tao theorem: an exposition, EMS Surv. Math. Sci. 1(2) (2014), 249282.CrossRefGoogle Scholar
3.Conlon, D., Fox, J. and Zhao, Y., A relative Szemerédi theorem, Geom. Funct. Anal. 25(3) (2015), 733762.CrossRefGoogle Scholar
4.Goldston, D. A., Pintz, J. and Yıldırım, C. Y., Primes in tuples. I, Ann. of Math. (2) 170(2) (2009), 819862.CrossRefGoogle Scholar
5.Gowers, W. T., A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11(3) (2001), 465588.CrossRefGoogle Scholar
6.Gowers, W. T., Decompositions, approximate structure, transference, and the Hahn–Banach theorem, Bull. London Math. Soc. 42(4) (2010), 573606.CrossRefGoogle Scholar
7.Green, B., Roth's theorem in the primes, Ann. of Math. (2) 161(3) (2005), 16091636.CrossRefGoogle Scholar
8.Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions, Ann. of Math. (2) 167(2) (2008), 481547.CrossRefGoogle 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), 767783.Google Scholar
11.Henriot, K., On systems of complexity one in the primes, Proc. Edinb. Math. Soc. (2) 60(1) (2016), 133163.CrossRefGoogle Scholar
12.Naslund, E., On improving Roth's theorem in the primes, Mathematika 61(1) (2015), 4962.CrossRefGoogle 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), 213305.CrossRefGoogle Scholar
16.Varnavides, P., On certain sets of positive density, J. Lond. Math. Soc. 34 (1959), 358360.CrossRefGoogle Scholar
17.Zhao, Y., An arithmetic transference proof of a relative Szemerédi theorem, Math. Proc. Cambridge Philos. Soc. 156(2) (2014), 255261.CrossRefGoogle Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Szemerédi's Theorem in the Primes
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Szemerédi's Theorem in the Primes
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Szemerédi's Theorem in the Primes
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *