|
A new proof of Roth's theorem on arithmetic progressions
Author(s):
Ernie
Croot;
Olof
Sisask
Journal:
Proc. Amer. Math. Soc.
137
(2009),
805-809.
MSC (2000):
Primary 05D99
Posted:
November 4, 2008
Retrieve article in:
PDF
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.
References:
-
- 1.
- J. Bourgain, On triples in arithmetic progression, Geom. and Funct. Anal. 9 (1999), 968-984. MR 1726234 (2001h:11132)
- 2.
- E. Croot, The structure of critical sets for
 arithmetic progressions, preprint. - 3.
- D.R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. 35 (1987), 385-394. MR 889362 (88g:11005)
- 4.
- K.F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953), 104-109. MR 0051853 (14:536g)
- 5.
- I. Z. Ruzsa and E. Szemerédi, Triple systems with no six points carrying three triangles, Combinatorics, Vol. II, pp. 939-945, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, 1978. MR 519318 (80c:05116)
- 6.
- E. Szemerédi, An old new proof of Roth's theorem, Montreal Conference Proceedings on Additive Combinatorics, CRM Proc. Lecture Notes, vol. 43, pp. 51-54, Amer. Math. Soc., Providence, RI, 2007. MR 2359467
- 7.
- -, Integer sets containing no arithmetic progressions, Acta Math. Hungar. 56 (1990), 155-158. MR 1100788 (92c:11100)
- 8.
- P. Varnavides, On certain sets of positive density, J. London Math. Soc. 34 (1959), 358-360. MR 0106865 (21:5595)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
05D99
Retrieve articles in all Journals with MSC
(2000):
05D99
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:
10.1090/S0002-9939-08-09594-4
PII:
S 0002-9939(08)09594-4
Received by editor(s):
January 17, 2008
Posted:
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 of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org