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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

New bounds on cap sets


Authors: Michael Bateman and Nets Hawk Katz
Journal: J. Amer. Math. Soc. 25 (2012), 585-613
MSC (2010): Primary 11T71; Secondary 05D40
Published electronically: November 29, 2011
MathSciNet review: 2869028
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Abstract: We provide an improvement over Meshulam's bound on cap sets in $ F_3^N$. We show that there exist universal $ \epsilon >0$ and $ C>0$ so that any cap set in $ F_3^N$ has size at most $ C {3^N \over N^{1+\epsilon }}$. We do this by obtaining quite strong information about the additive combinatorial properties of the large spectrum.


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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 47405-7000
Email: nhkatz@indiana.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2011-00725-X
PII: S 0894-0347(2011)00725-X
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, DMS-0902490
The second author is partially supported by NSF grant DMS-1001607
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.