IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

NSF-CBMS Conference on Additive Combinatorics from a Geometric Viewpoint

Joshua Zahl
University of British Columbia
https://www.math.ubc.ca/~jzahl/
Abstract

The discretized sum-product problem

  • May 25, 2018
  • 10:30 a.m.
  • LeConte 412

Classical sum-product 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 "r-covering 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 sum-product theorem. I will discuss the discretized sum-product theorem, as well as some recent developments.

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