Math Graduate Student Seminar

The Roth theorem is a result in combinatorics about finding three-term arithmetic progressions in a large subset of integers, and the Polynomial Szemeredi theorem is the generalization about finding certain polynomial progressions. Motivated by the questions related to ergodic theory, Bourgain and Chang initiated the study of the Polynomial Szemeredi-type questions in the finite field setting in 2016 and left some interesting directions to be explored.

In this talk, I will present a recent result showing that the three-term rational progression can always be found for large sets in the finite field setting, which generalized Bourgain and Chang's previous result for a specific configuration: x, x+y, x+1/y. At the end of the talk, I will discuss further explorations in the continuous setting.

This is a joint work with Zi Li Lim at UCLA.