  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 $|A|\geq (c_q q)^n$ 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, L\'aszl\'o
Mikl\'os Lov\'asz, A tight bound for Green's arithmetic triangle removal lemma
in vector spaces, preprint, arXiv:1606.01230.
