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 M. Korobov (1963) proved that one can take M = O(\log p), and in 2022 Moshchevitin--Murphy--Shkredov used the growth in groups 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}.
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