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arXiv · 1708.08482

Popular progression differences in vector spaces

Jacob Fox·Huy Tuan Pham
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Abstract

Green proved an arithmetic analogue of Szemer\'edi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that for every subset of $\mathbb{F}_p^n$ with $n$ sufficiently large, the density of three-term arithmetic progressions with some nonzero common difference is at least the random bound (the cube of the set density) up to an additive $\epsilon$. For a fixed odd prime $p$, we prove that the required dimension grows as an exponential tower of $p$'s of height $\Theta(\log(1/\epsilon))$. This improves both the lower and upper bound, and is the first example of a result where a tower-type bound coming from applying a regularity lemma is shown to be necessary.

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Jacob Fox, Huy Tuan Pham. 2017-08-28. Popular progression differences in vector spaces. https://arxiv.org/abs/1708.08482

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