New Bounds on cap sets
By M. Bateman, Nets Hawk Katz. 2011
99 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/ef3c0db3638a504250d05e02073f36ec7c30f8f3
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In Computational Theory and Mathematics
In Discrete Mathematics and Combinatorics
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Citations in the corpus
- On large subsets of F nq with no three-termarithmetic progression (2017, unknown): “We know ( instance, by a construction of Behrend [Beh46]) that r 3 ( Z /N Z ) grows more quickly than N for every ε> 0.”
- Progression-free sets in Z_4^n are exponentially small (2016, background): “use the standard notation rk(G) for the rank of G); in particular, r3(Zn m) ≤ 2mn/n. Despitemany efforts, nofurtherprogress wasmade forover 15years, tillBateman and Katz in their ground-breaking paper [BK12] proved that r3(Zn 3) = O(3n/n1+ε) with an absolute constant ε > 0. Abelian groups of even ”