On Korobov bound concerning Zaremba's conjecture

TL;DR

This paper proves that for large primes, there exists a fraction with denominator p whose partial quotients are bounded by a logarithmic function, improving the Korobov bound related to Zaremba's conjecture.

Contribution

It extends the Korobov bound to composite denominators, providing a tighter bound on partial quotients for fractions related to Zaremba's conjecture.

Findings

Existence of fractions with bounded partial quotients for large primes

Extension of bounds to composite denominators

Improvement over previous Korobov bounds

Abstract

We prove in particular that for any sufficiently large prime $p$ there is $1\le a<p$ such that all partial quotients of $a/p$ are bounded by $O(\log p/\log \log p)$. For composite denominators a similar result is obtained. This improves the well--known Korobov bound concerning Zaremba's conjecture from the theory of continued fractions.