{"paper":{"title":"On some results of Korobov and Larcher and Zaremba's conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"Ilya D. 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 $O(\\log q \\cdot \\sqrt{\\log \\log q})$. This, accordingly, imp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.14116","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.14116/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}