NSFCBMS Conference on Additive Combinatorics from a Geometric Viewpoint
Joshua Zahl
University of British Columbia https://www.math.ubc.ca/~jzahl/ 

Abstract 
The discretized sumproduct problem
Classical sumproduct theory asserts that if A is a finite set of real numbers, then either A+A or A.A must have cardinality much later than that of A. In 2001, Katz and Tao conjectured that a similar statement should be true if "cardinality" is replaced by "rcovering number": Given a set A of real numbers (satisfying certain hypotheses), and a small number r>0, the number of intervals of length r required to cover either A+A or A.A must be much larger than the number of intervals required to cover A. Katz and Tao's conjecture is closely related to several problems in harmonic analysis. Their conjecture was resolved by Bourgain, who proved what is now called the discretized sumproduct theorem. I will discuss the discretized sumproduct theorem, as well as some recent developments. 