On a modular form of Zaremba's conjecture

TL;DR

This paper proves the existence of fractions with denominators divisible by any prime p that have bounded partial quotients, advancing understanding of Zaremba's conjecture and related number theory problems.

Contribution

It establishes bounds on denominators divisible by primes for fractions with bounded partial quotients, providing new results in the study of Zaremba's conjecture.

Findings

Existence of denominators divisible by p with bounded partial quotients

Bounded partial quotients with denominators of size O(p^{30})

Universal constant C for denominators with partial quotients bounded by two

Abstract

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.