On some results of Korobov and Larcher and Zaremba's conjecture

TL;DR

This paper proves the Zaremba conjecture for prime denominators, showing that for large q, there are many coprime a with bounded partial quotients, improving previous bounds and providing new asymptotic estimates.

Contribution

It establishes new bounds on partial quotients in continued fractions for prime denominators, confirming Zaremba's conjecture in this case and improving earlier results.

Findings

Existence of a coprime a with bounded partial quotients for large q

Asymptotically tight lower bounds for such a

Improved bounds on the sum of partial quotients

Abstract

We prove, in particular, the well--known Zaremba conjecture from the theory of continued fractions for any prime denominator. More precisely, we show, firstly, that under some mild conditions, for any sufficiently large $q$, there exists $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $O(\sqrt{\log q})$, and, moreover we find asymptotically tight lower bound for the number of such $a$. Secondly, we obtain a good lower bound for the number $a$ such that the sum of all partial quotients of $a/q$ is bounded by $O(\log q \cdot \sqrt{\log \log q})$. This, accordingly, improves on some results of Korobov and Larcher. Finally, we show that for all sufficiently large $\mathcal{M}$ there are $\Omega(q^{1-O(1/\mathcal{M})})$ numbers $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $\mathcal{M}$.