Abstract Á Jean Bourgain avec admiration et tristesse. We prove in particular that for any sufficiently large prime p there is 1 ap such that all partial quotients of a/p are bounded by O (p/ p). For composite denominators a similar result is obtained. This improves Korobov’s O (p) bound, known since the 1960s, for Zaremba’s conjecture in continued fraction theory.
Moshchevitin et al. (2026) studied this question.