Green proved the following strengthening of Roth’s theorem: for every positive ϵ, there is some n(ε) such that for every N ≥ n(ϵ) and A ⊂ [N] with |A| = αN, there is some nonzero d such that A contains at least (α3 − ϵ)N three-term arithmetic progressions with common difference d (i.e., a popular common difference with frequently at least roughly the random bound). I’ll discuss some extensions and generalizations of this result:
- How large does n(ϵ) have to be for the result to hold? (It turns out that a tower-type bound is necessary)
- Besides 3-term arithmetic progressions, is there a similar result for other patterns?
- What about patterns in higher-dimensional patterns?
Based on joint works with Jacob Fox, Huy Tuan Pham, Ashwin Sah, Mehtaab Sawhney, and David Stoner.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
