On a modular form of Zaremba’s conjecture

We prove that for any prime $p$ there is a divisible by $p$ number $q = O (p^{30})$ 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 $q = O (p^{C})$ and a number $a$ , $a$ coprime with $q$ such that all partial quotients of the ratio $a ∕ q$ are bounded by two.

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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

Authors

Nikolay G. Moshchevitin
Steklov Mathematical Institute Moscow Russia

Ilya D. Shkredov
Steklov Mathematical Institute Moscow Russia

Vol. 309, No. 1, 2020

Nikolay G. Moshchevitin and Ilya D. Shkredov

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors