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

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 $Ω(q^{1-O(1/\mathcal{M})})$ numbers $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $\mathcal{M}$.