Ilya D. Shkredov: Zaremba's conjecture and growth in groups (NTWS 036)
Channel: Number Theory Web Seminar
Date: 2023-06-04

Abstract: Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a less than q, (a,q)=1 such that all  partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +...  + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension strictly greater than 1/2 takes place for the so-called modular form of Zaremba's conjecture.  

Link to slides: https://drive.google.com/file/d/1Jw-NWBfL29qqaWdXks4gUvLPSYRQHEzi

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

Original air date: 
Tuesday, September 22, 2020 (2am PDT, 5am EDT, 10am BST, 11am CEST, 12pm IDT, 2:30pm IST, 5pm China Standard Time, 7pm AEST, 9pm NZST)