# Seminars & Colloquia Calendar

## 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.

Chiara Damiolini, Ian Coley and Franco Rota -Charles Weibel Organizer's Page

Brooke Logan

Wujun Zhang Organizer's webpage

Ziming Shi, Sagun Chanillo, Xiaojun Huang, Chi Li, Jian Song Seminar website Old seminar website

Swastik Kopparty, Sepehr Assadi Seminar webpage

Jeffry Kahn, Bhargav Narayanan, Jinyoung Park Organizer's webpage

Brooke Ogrodnik, Website

Robert Dougherty-Bliss and Doron Zeilberger --> homepage

Paul Feehan, Daniel Ketover, Natasa Sesum Organizer's webpage

Lev Borisov, Emanuel Diaconescu, Angela Gibney, Nicolas Tarasca, and Chris Woodward Organizer's webpage

Jason Saied Seminar webpage

Brian Pinsky, Rashmika Goswami website

Quentin Dubroff Organizer's webpage

James Holland; Organizer website

Edna Jones Organizer's webpage

Brooke Ogrodnik website

Yanyan Li, Zheng-Chao Han, Jian Song, Natasa Sesum Organizer's Webpage

Organizer: Luochen Zhao

Yanyan Li, Zheng-Chao Han, Natasa Sesum, Jian Song Organizer's Page

Lisa Carbone, Yi-Zhi Huang, James Lepowsky, Siddhartha Sahi Organizer's webpage

Simon Thomas website

Kasper Larsen, Daniel Ocone and Kim Weston Organizer's page

Joel Lebowitz, Michael Kiessling

Yanyan Li, Haim Brezis Organizer's Webpage

Stephen D. Miller, John C. Miller, Alex V. Kontorovich, Alex Walker seminar website

Stephen D. Miller

Brooke Ogrodnik, Website

Organizers: Yanyan Li, Z.C. Han, Jian Song, Natasa Sesum

Yael Davidov Seminar webpage

Kristen Hendricks, Xiaochun Rong, Hongbin Sun, Chenxi Wu Organizer's page

Fioralba Cakoni Seminar webpage

Ebru Toprak, Organizer

- Show events from all categories

## Special Note to All Travelers

Directions: map and driving directions. If you need information on public transportation, you may want to check the New Jersey Transit page.

*Unfortunately, cancellations do occur from time to time. Feel free to call our department: 848-445-6969 before embarking on your journey. Thank you.*