A Polynomial Roth Theorem for Corners in the Finite Field Setting

Analysis Seminar
Wednesday, January 27, 2021 - 2:00pm for 1 hour (actually 50 minutes)
Michael Lacey – Georgia Tech
Benjamin Jaye

An initial result of Bourgain and Chang has lead to a number of striking advances in the understanding of polynomial extensions of Roth's Theorem.
The most striking of these is the result of Peluse and Prendiville which show that sets in [1 ,..., N] with density greater than (\log N)^{-c} contain polynomial progressions of length k (where c=c(k)).  There is as of yet no corresponding result for corners, the two dimensional setting for Roth's Theorem, where one would seek progressions of the form(x,y), (x+t^2, y), (x,y+t^3) in  [1 ,..., N]^2, for example.  

Recently, the corners version of the result of Bourgain and Chang has been established, showing an effective bound for a three term polynomial Roth theorem in the finite field setting.  We will survey this area. Joint work with Rui Han and Fan Yang.

The link for the seminar is the following