On Korobov Bound Concerning Zaremba’s Conjecture

Abstract Á Jean Bourgain avec admiration et tristesse. We prove in particular that for any sufficiently large prime $p$ there is $1\leqslant a<p$ 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 Korobov’s $O(\log p)$ bound, known since the 1960s, for Zaremba’s conjecture in continued fraction theory.