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Shkredov","submitted_at":"2026-03-14T21:05:48Z","abstract_excerpt":"We prove, in particular, the well--known Zaremba conjecture from the theory of continued fractions for any prime denominator. More precisely, we show, firstly, that under some mild conditions, for any sufficiently large $q$, there exists $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $O(\\sqrt{\\log q})$, and, moreover we find asymptotically tight lower bound for the number of such $a$. Secondly, we obtain a good lower bound for the number $a$ such that the sum of all partial quotients of $a/q$ is bounded by $O(\\log q \\cdot \\sqrt{\\log \\log q})$. 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