On the number of three-term arithmetic progressions in a dense subset of $F_q^n$

Abstract

Let $q$ be an odd prime power. Combining the discussion of Varnavides and a recent theorem of Ellenberg and Gijswijt, we show that a subset $A\subset {\mathbb{F}}_q^n$ will contain many non-trivial three-term arithmetic progressions, whenever $|A|\ge (c_{q}q{)}^n$ for some constant $c_q>0$. After the first version of our manuscript was uploaded in the arXiv, we learned from Professors Jacob Fox and Terence Tao that our result is a special case of a result of Fox and Lovasz [1, Theorem 3]. In fact, [1, Theorem 3] gives a much better bound than ours. For example, when $q=3$, the lower bound given by Fox and Lovasz is $|A{|}^2\cdot (|A|q^{-n}{)}^{11.901}$, while our bound is $|A{|}^2\cdot (|A|q^{-n}{)}^{25.803}$. We thank Professors Jacob Fox and Terence Tao for their helpful comments on our manuscript. [1] Jacob Fox, László Miklós Lovász, A tight bound for Green's arithmetic triangle removal lemma in vector spaces, preprint, arXiv:1606.01230.

Our result is weaker than a known result of Fox and Lovasz