Vol. 309, No. 1, 2020

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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(p30) such that for a certain positive integer a coprime with q the ratio aq 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(pC) and a number a, a coprime with q such that all partial quotients of the ratio aq 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
Authors
Nikolay G. Moshchevitin
Steklov Mathematical Institute
Moscow
Russia
Ilya D. Shkredov
Steklov Mathematical Institute
Moscow
Russia