Bourgain: additive and multiplicative structure of finite sets
Statement
A landmark line of work (initiated by Bourgain and developed with many coauthors) shows that finite sets of real numbers (or elements in other rings) cannot simultaneously have minimal additive and multiplicative structure: quantitative lower bounds on max(|A+A|, |A·A|) and related energies underlie applications in combinatorics, harmonic analysis, and number theory. Representative early paper: sum-product estimate in ℝ. Proved in ZFC.
Source
J. Bourgain, N. Katz, T. Tao, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), 27–57; broader additive-combinatorics program by Bourgain and collaborators.
Domains
mathematics
People
Related
- Theorem: Bourgain: additive and multiplicative structure of finite sets
- Proof: Bourgain: additive and multiplicative structure of finite sets