# Seminars & Colloquia Calendar

Discrete Math

## Tower-type bounds for Roth's theorem with popular differences

#### Yufei Zhao, MIT

Location:  Hill 705
Date & time: Monday, 16 April 2018 at 2:00PM - 3:00PM

Abstract: A famous theorem of Roth states that for any $$\alpha > 0$$ and $$n$$ sufficiently large in terms of $$\alpha$$, any subset of $$\{1, dots, n\}$$ with density $$\alpha$$ contains a 3-term arithmetic progression. Green developed an arithmetic regularity lemma and used it to prove that not only is there one arithmetic progression, but in fact there is some integer $$d > 0$$ for which the density of 3-term arithmetic progressions with common difference $$d$$ is at least roughly what is expected in a random set with density $$\alpha$$. That is, for every $$\epsilon > 0$$, there is some $$n(\epsilon)$$ such that for all $$n > n(\epsilon)$$ and any subset $$A$$ of $$\{1, dots, n\}$$ with density $$\alpha$$, there is some integer $$d > 0$$ for which the number of 3-term arithmetic progressions in $$A$$ with common difference $$d$$ is at least $$(\alpha^3-\epsilon)n$$. We prove that $$n(\epsilon)$$ grows as an exponential tower of 2's of height on the order of $$\log(1/\epsilon)$$. We show that the same is true in any abelian group of odd order $$n$$. These results are the first applications of regularity lemmas for which the tower-type bounds are shown to be necessary.

Joint work with Jacob Fox and Huy Tuan Pham.

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