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- W2556556877 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 $Asubset{mathbb F}_q^n$ 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, Laszlo Miklos Lovasz, A tight bound for Green's arithmetic triangle removal lemma in vector spaces, preprint, arXiv:1606.01230." @default.
- W2556556877 created "2016-11-30" @default.
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- W2556556877 date "2016-11-23" @default.
- W2556556877 modified "2023-09-27" @default.
- W2556556877 title "On the number of three-term arithmetic progressions in a dense subset of $F_q^n$" @default.
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