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

TL;DR

This paper demonstrates that dense subsets of finite vector spaces over finite fields contain many three-term arithmetic progressions, using combinatorial and algebraic methods, and discusses how their bounds compare to existing results.

Contribution

The authors provide a bound on the number of three-term arithmetic progressions in dense subsets of finite fields, connecting combinatorial and algebraic techniques, and relate their results to recent theorems.

Findings

Dense subsets contain many 3-term arithmetic progressions

Established bounds on the number of progressions in terms of subset density

Compared bounds with recent results, noting differences in tightness

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 $|A|\geq (c_q q)^n$ 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.