On a modular form of Zaremba's conjecture

Nikolay G. Moshchevitin and Ilya D. Shkredov

Vol. 309 (2020), No. 1, 195–211

DOI: 10.2140/pjm.2020.309.195

Abstract

We prove that for any prime $p$ there is a divisible by $p$ number $q=O\left(p^{30}\right)$ such that for a certain positive integer $a$ coprime with $q$ the ratio $a∕q$ has bounded partial quotients. In the other direction we show that there is an absolute constant $C>0$ such that for any prime $p$ exist divisible by $p$ number $q=O\left(p^C\right)$ and a number $a$, $a$ coprime with $q$ such that all partial quotients of the ratio $a∕q$ are bounded by two.

Keywords

continued fractions, Zaremba's conjecture, growth in groups

Mathematical Subject Classification

Primary: 11B13, 11B75, 11E57, 11J70

Milestones

Received: 18 November 2019

Revised: 12 August 2020

Accepted: 1 September 2020

Published: 26 December 2020