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

41 pages. 1. Some minor typos have been corrected, and clarifications have been made. 2. For the reader's convenience, the proof of technical Lemma 32 has been broken down into three auxiliary lemmas, simplified, and clarified. 3. In the appendix, we provide a crude upper bound for the absolute constant $\mathcal{M}$ in the case of a prime denominator