A new proof of Roth’s theorem on arithmetic progressions
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- by Ernie Croot and Olof Sisask PDF
- Proc. Amer. Math. Soc. 137 (2009), 805-809 Request permission
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.References
- J. Bourgain, On triples in arithmetic progression, Geom. Funct. Anal. 9 (1999), no. 5, 968–984. MR 1726234, DOI 10.1007/s000390050105
- E. Croot, The structure of critical sets for $\mathbb {F}_p$ arithmetic progressions, preprint.
- D. R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. (2) 35 (1987), no. 3, 385–394. MR 889362, DOI 10.1112/jlms/s2-35.3.385
- K. F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109. MR 51853, DOI 10.1112/jlms/s1-28.1.104
- I. Z. Ruzsa and E. Szemerédi, Triple systems with no six points carrying three triangles, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976) Colloq. Math. Soc. János Bolyai, vol. 18, North-Holland, Amsterdam-New York, 1978, pp. 939–945. MR 519318
- Endre Szemerédi, An old new proof of Roth’s theorem, Additive combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 51–54. MR 2359467, DOI 10.1090/crmp/043/04
- E. Szemerédi, Integer sets containing no arithmetic progressions, Acta Math. Hungar. 56 (1990), no. 1-2, 155–158. MR 1100788, DOI 10.1007/BF01903717
- P. Varnavides, On certain sets of positive density, J. London Math. Soc. 34 (1959), 358–360. MR 106865, DOI 10.1112/jlms/s1-34.3.358
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
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 805-809
- MSC (2000): Primary 05D99
- DOI: https://doi.org/10.1090/S0002-9939-08-09594-4
- MathSciNet review: 2457417