New bounds on cap sets
Authors:
Michael Bateman and Nets Hawk Katz
Journal:
J. Amer. Math. Soc. 25 (2012), 585613
MSC (2010):
Primary 11T71; Secondary 05D40
Published electronically:
November 29, 2011
MathSciNet review:
2869028
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We provide an improvement over Meshulam's bound on cap sets in . We show that there exist universal and so that any cap set in has size at most . We do this by obtaining quite strong information about the additive combinatorial properties of the large spectrum.
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 E. Croot and O. Sisask, A Probabilistic Technique for Finding AlmostPeriods of Convolutions, Geom. Funct. Anal. 20 (2010), 13671396. MR 2738997
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 T. Gowers, What is difficult about the cap set problem?, http://gowers.wordpress.com/ 2011/01/11/whatisdifficultaboutthecapsetproblem/.
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 (M. Bateman and) N.H. Katz, http://gowers.wordpress.com/2011/01/11/whatisdifficultaboutthecapsetproblem/# comment10533.
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 N. H. Katz and P. Koester, On Additive Doubling and Energy, SIAM J. Discrete Math. 24 (2010), 16841693. MR 2746716
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 R. Meshulam, On subsets of finite abelian groups with no term arithmetic progressions, J. Combin. Theory Ser. A 71 (1995), 168172. MR 1335785 (96g:20033)
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 I. Ruzsa, An analog of Freiman's theorem in groups, Structure theory of set addition, Astérisque No. 258 (1999), 323329. MR 1701207 (2000h:11111)
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 T. Sanders, A note on Freiman's theorem in vector spaces, Combin. Probab. Comput. 17 (2008), 297305. MR 2396355 (2009a:11024)
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 T. Sanders, Structure in Sets with Logarithmic Doubling, Arxiv 1002.1552.
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 T. Sanders, On Roth's Theorem on Progressions, Arxiv 1011.0104.
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 T. Schoen, Near optimal bounds in Freiman's theorem, Duke Math. J. 158, Number 1 (2011), 112. MR 2794366
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 I. Shkredov, On sets of large trigonometric sums, 2008 Izv. Ross. Akad. Nauk Ser. Math. 72, pp. 161182. MR 2394976 (2009e:11151)
 [TV06]
 T. Tao and V. Vu, Additive Combinatorics, Cambridge Univ. Press, Cambridge, 2006. MR 2289012 (2008a:11002)
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Additional Information
Michael Bateman
Affiliation:
Department of Mathematics, UCLA, Los Angeles, California 90095
Email:
bateman@math.ucla.edu
Nets Hawk Katz
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 474057000
Email:
nhkatz@indiana.edu
DOI:
http://dx.doi.org/10.1090/S08940347201100725X
Received by editor(s):
April 2, 2011
Received by editor(s) in revised form:
October 28, 2011
Published electronically:
November 29, 2011
Additional Notes:
The first author is supported by an NSF postdoctoral fellowship, DMS0902490
The second author is partially supported by NSF grant DMS1001607
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
