#### 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 $q=O\left({p}^{30}\right)$ 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