On large subsets of $F_q^n$ with no three-term arithmetic progression

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 $F_q^n$ with no three terms in arithmetic progression by $c^n$ with $c<q$. For $q=3$, the problem of finding the largest subset with no three terms in arithmetic progression is called the `cap problem'. Previously the best known upper bound for the cap problem, due to Bateman and Katz, was $O(3^n/n^{1+\epsilon })$.

4 pages. This paper supersedes arXiv:1605.05492 and combines the solutions to the cap set problem independently obtained by the two authors

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