On large subsets of F<i>nq</i> with no three-termarithmetic progression

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&lt;q c&lt;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