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

DOI:
https://doi.org/10.1090/S0002-9939-08-09594-4

Published electronically:
November 4, 2008

MathSciNet review:
2457417

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-08-09594-4

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.