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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A new proof of Roth's theorem on arithmetic progressions


Authors: Ernie Croot and Olof Sisask
Journal: Proc. Amer. Math. Soc. 137 (2009), 805-809
MSC (2000): Primary 05D99
Published electronically: November 4, 2008
MathSciNet review: 2457417
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Abstract: We present a proof of Roth's theorem that follows a slightly different structure to the usual proofs, in that there is not much iteration. Although our proof works using a type of density increment argument (which is typical of most proofs of Roth's theorem), we do not pass to a progression related to the large Fourier coefficients of our set (as most other proofs of Roth do). Furthermore, in our proof, the density increment is achieved through an application of a quantitative version of Varnavides's theorem, which is perhaps unexpected.


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Additional Information

Ernie Croot
Affiliation: Department of Mathematics, Georgia Institute of Technology, 103 Skiles, Atlanta, Georgia 30332
Email: ecroot@math.gatech.edu

Olof Sisask
Affiliation: Department of Mathematics, University of Bristol, Bristol BS8 1TW, England
Email: O.Sisask@dpmms.cam.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09594-4
PII: S 0002-9939(08)09594-4
Received by editor(s): January 17, 2008
Published electronically: November 4, 2008
Additional Notes: The first author was funded by NSF grant DMS-0500863.
The second author was funded by an EPSRC DTG through the University of Bristol, and he would like to thank the University of Cambridge for its kind hospitality while this work was carried out.
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.