## New bounds on cap sets

HTML articles powered by AMS MathViewer

- by Michael Bateman and Nets Hawk Katz PDF
- J. Amer. Math. Soc.
**25**(2012), 585-613 Request permission

## 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.## References

- Ernie Croot and Olof Sisask,
*A probabilistic technique for finding almost-periods of convolutions*, Geom. Funct. Anal.**20**(2010), no. 6, 1367–1396. MR**2738997**, DOI 10.1007/s00039-010-0101-8 - T. Gowers,
*What is difficult about the cap set problem?*, http://gowers.wordpress.com/ 2011/01/11/what-is-difficult-about-the-cap-set-problem/. - (M. Bateman and) N.H. Katz, http://gowers.wordpress.com/2011/01/11/what-is-difficult-about-the-cap-set-problem/# comment-10533.
- (M. Bateman and) N.H. Katz, http://gowers.wordpress.com/2011/01/11/what-is-difficult-about-the-cap-set-problem/# comment-10540.
- Nets Hawk Katz and Paul Koester,
*On additive doubling and energy*, SIAM J. Discrete Math.**24**(2010), no. 4, 1684–1693. MR**2746716**, DOI 10.1137/080717286 - Roy Meshulam,
*On subsets of finite abelian groups with no $3$-term arithmetic progressions*, J. Combin. Theory Ser. A**71**(1995), no. 1, 168–172. MR**1335785**, DOI 10.1016/0097-3165(95)90024-1 - Polymath on wikipedia, http://en.wikipedia.org/wiki/Polymath_project#Polymath _Project.
- Polymath 6: Improving the bounds for Roth’s theorem, http://polymathprojects.org/ 2011/02/05/polymath6-improving-the-bounds-for-roths-theorem/.
- Imre Z. Ruzsa,
*An analog of Freiman’s theorem in groups*, Astérisque**258**(1999), xv, 323–326 (English, with English and French summaries). Structure theory of set addition. MR**1701207** - T. Sanders,
*A note on Freĭman’s theorem in vector spaces*, Combin. Probab. Comput.**17**(2008), no. 2, 297–305. MR**2396355**, DOI 10.1017/S0963548307008644 - T. Sanders,
*Structure in Sets with Logarithmic Doubling*, Arxiv 1002.1552. - T. Sanders,
*On Roth’s Theorem on Progressions*, Arxiv 1011.0104. - Tomasz Schoen,
*Near optimal bounds in Freiman’s theorem*, Duke Math. J.**158**(2011), no. 1, 1–12. MR**2794366**, DOI 10.1215/00127094-1276283 - I. D. Shkredov,
*On sets of large trigonometric sums*, Izv. Ross. Akad. Nauk Ser. Mat.**72**(2008), no. 1, 161–182 (Russian, with Russian summary); English transl., Izv. Math.**72**(2008), no. 1, 149–168. MR**2394976**, DOI 10.1070/IM2008v072n01ABEH002396 - Terence Tao and Van Vu,
*Additive combinatorics*, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR**2289012**, DOI 10.1017/CBO9780511755149

## 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
- MR Author ID: 610432
- Email: nhkatz@indiana.edu
- 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 - © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**25**(2012), 585-613 - MSC (2010): Primary 11T71; Secondary 05D40
- DOI: https://doi.org/10.1090/S0894-0347-2011-00725-X
- MathSciNet review: 2869028