Misha Rudnev: The sum-product problem for integers with few prime factors (NTWS 192)
Channel: Number Theory Web Seminar
Date: 2023-12-11

Abstract: It was asked by Szemerédi if the known sum-product estimates can be improved for a set of N integers under the constraint that each integer has a small number of prime factors. We prove, if the maximum number of prime factors for each integer is sub-logarithmic in N, the sum-product exponent 5/3-o(1). 
This becomes a corollary of an additive energy versus the product set cardinality estimate, which turns out to be the best possible. 
It is based on a scheme of Burkholder-Gundy-Davis martingale square function inequalities in p-adic scales, followed by an application of a variant of the Schmidt subspace theorem.

Link to slides: https://drive.google.com/file/d/156TmROiC2yw4-FodlfOhsDgWTVzaj7mL

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, December 7, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, December 8, 2023 (12am CST, 3am AEDT, 5am NZDT)