Yuyan He : Sum product phenomenon in fields with dimension

Posted by Gabriel Conant , part of the Logic Seminar.

At

March 3, 2026, 3 p.m.

In

636 SEO

Abstract

A classical result of Ruzsa and Szemeredi shows that there is no subset A of the integers such that both the sumset A+A and the productset AA are small. Breuillard, Katz and Tao found a similar result for the subsets of finite prime fields of “medium size”. We show that under certain mild assumptions on a dimension theory , a “medium sized” subset of a field with small sumset and productset in the sense of not expanding in dimension indicates the existence of a subfield of the same dimension. This is joint work with Sergei Starchenko.