Popular progression differences in vector spaces II

Jacob Fox ; Huy Tuan Pham

Discrete analysis (2019)

Résumé

Green used an arithmetic analogue of Szemerédi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every $α>0$, $β$, and prime number $p$, there is a least positive integer $n_p(α,β)$ such that if $n \geq n_p(α,β)$, then for every subset of $\mathbb{F}_p^n$ of density at least $α$ there is a nonzero $d$ for which the density of three-term arithmetic progressions with common difference $d$ is at least $β$. We determine for $p \geq 19$ the tower height of $n_p(α,β)$ up to an absolute constant factor and an additive term depending only on $p$. In particular, if we want half the random bound (so $β=α^3/2$), then the dimension $n$ required is a tower of twos of height $Θ\left((\log p) \log \log (1/α)\right)$. It turns out that the tower height in general takes on a different form in several different regions of $α$ and $β$, and different arguments are used both in the upper and lower bounds to handle these cases.

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Publié le : 2019-11-20