On some results of Korobov and Larcher and Zaremba's conjecture
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914393647742976 |
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| author | Shkredov, Ilya D. |
| author_facet | Shkredov, Ilya D. |
| contents | 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 $O(\log q \cdot \sqrt{\log \log q})$. This, accordingly, improves on some results of Korobov and Larcher. Finally, we show that for all sufficiently large $\mathcal{M}$ there are $Ω(q^{1-O(1/\mathcal{M})})$ numbers $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $\mathcal{M}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14116 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On some results of Korobov and Larcher and Zaremba's conjecture Shkredov, Ilya D. Number Theory Classical Analysis and ODEs Combinatorics 11J70, 11B30, 05C25, 20G40, 11B75 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 $O(\log q \cdot \sqrt{\log \log q})$. This, accordingly, improves on some results of Korobov and Larcher. Finally, we show that for all sufficiently large $\mathcal{M}$ there are $Ω(q^{1-O(1/\mathcal{M})})$ numbers $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $\mathcal{M}$. |
| title | On some results of Korobov and Larcher and Zaremba's conjecture |
| topic | Number Theory Classical Analysis and ODEs Combinatorics 11J70, 11B30, 05C25, 20G40, 11B75 |
| url | https://arxiv.org/abs/2603.14116 |