On large subsets of Fnq with no three-termarithmetic progression
Jordan S. Ellenberg (University of Wisconsin-Madison)
Dion Gijswijt (TU Delft - Discrete Mathematics and Optimization)
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Abstract
In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of F n q Fqn
with no three terms in arithmetic progression by c n cn
with c<q c<q
. For q=3 q=3
, the problem of finding the largest subset of F n 3 F3n
with no three terms in arithmetic progression is called the cap set problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz, was on order n −1−ϵ 3 n n−1−ϵ3n
.