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New bounds on cap sets. (English) Zbl 1262.11010
A set $$A\subseteq \mathbb F^n_3$$ is called a cap set if it contains no solutions of the equation $$x+ y+ 7= 0$$, $$x,y,7\in A$$, $$x\neq y\neq 7$$. These sets correspond to arithmetic progressions of the length three in $$\mathbb Z$$. Improving a result of R. Meshulom the authors prove that any cap set has size less than $$O({3^n\over n^{1+\varepsilon}})$$, where $$\varepsilon> 0$$ is an absolute (but small) constant. It corresponds to the famous conjecture of Erdős and Turán on sets without arithmetic progressions. The proof based on new structural results about sets with a nontrivial lower bounds for the additive energy as well as new statements on the spetrum of cap sets.

##### MSC:
 11B30 Arithmetic combinatorics; higher degree uniformity 11T99 Finite fields and commutative rings (number-theoretic aspects) 05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.) 11B75 Other combinatorial number theory
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##### References:
 [1] Ernie Croot and Olof Sisask, A probabilistic technique for finding almost-periods of convolutions, Geom. Funct. Anal. 20 (2010), no. 6, 1367 – 1396. · Zbl 1234.11013 [2] 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/. [3] (M. Bateman and) N.H. Katz, http://gowers.wordpress.com/2011/01/11/what-is-difficult-about-the-cap-set-problem/# comment-10533. [4] (M. Bateman and) N.H. Katz, http://gowers.wordpress.com/2011/01/11/what-is-difficult-about-the-cap-set-problem/# comment-10540. [5] Nets Hawk Katz and Paul Koester, On additive doubling and energy, SIAM J. Discrete Math. 24 (2010), no. 4, 1684 – 1693. · Zbl 1226.05247 [6] 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. · Zbl 0832.11006 [7] Polymath on wikipedia, http://en.wikipedia.org/wiki/Polymath_project#Polymath _Project. [8] Polymath 6: Improving the bounds for Roth’s theorem, http://polymathprojects.org/ 2011/02/05/polymath6-improving-the-bounds-for-roths-theorem/. [9] 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. · Zbl 0946.11007 [10] T. Sanders, A note on Freĭman’s theorem in vector spaces, Combin. Probab. Comput. 17 (2008), no. 2, 297 – 305. · Zbl 1151.15003 [11] T. Sanders, Structure in Sets with Logarithmic Doubling, Arxiv 1002.1552. · Zbl 1310.11013 [12] T. Sanders, On Roth’s Theorem on Progressions, Arxiv 1011.0104. · Zbl 1264.11004 [13] Tomasz Schoen, Near optimal bounds in Freiman’s theorem, Duke Math. J. 158 (2011), no. 1, 1 – 12. · Zbl 1242.11074 [14] 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. · Zbl 1148.11040 [15] Terence Tao and Van Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. · Zbl 1127.11002
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