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

Zaremba's famous conjecture (1972) arose from the theory of numerical integration and relates to the field of continued fractions. It predicts that for any given prime p there is a positive integer a < p such that when expanded as a continued fraction a/p = 1/c_1+1/c_2 +... + 1/c_s all partial quotients c_j are bounded by a constant N. Korobov (1963) proved that one can take M = O(\log p), and in 2022 Moshchevitin--Murphy--Shkredov used the growth in SL_2 (\Z/p\Z) and multiplicative combinatorics to obtain that M=O(\log p/\log \log p). 

By applying some additional Diophantine and combinatorial ideas to the distribution of so-called critical denominators, we confirm Zaremba's conjecture for primes and for composite p satisfying some mild conditions. Moreover, we show that the number of fractions a/p such that c_j \le M is equal to \Omega(p^{1-O(1/M)}), confirming Hensley's heuristic. Finally, by studying continued fractions a/p = 1/c_1 + 1/c_2 + ... + 1/c_s with c_j \le M, where M \in [1, \log p] is a parameter, we discovered an interesting new threshold M = \sqrt {\log p}. 

Since the growth theory in SL_2 (\Z/q\Z) is now known for all q (see J. Tang--X. Zhang, 2023, 2025), this implies that Zaremba's conjecture takes place in its full generality.