Zaremba's conjecture and growth in groups

Speaker: 

Ilya D. Shkredov

Affiliation: 

Steklov Mathematical Institute, Moscow

Date: 

Tue, 22/09/2020 - 7:00pm

Venue: 

RC-4082, The Red Centre, UNSW

Abstract: 

Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer $q$ there is a positive $a$, $a < q$, $(a,q)=1$ such that allĀ  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 $>1/2$ takes place for the so-called modular form of Zaremba's conjecture.

This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.

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Organisers:
Mike Bennett (University of British Columbia)
Philipp Habegger (University of Basel)
Alina Ostafe (UNSW Sydney)

School Seminar Series: