{"paper":{"title":"On the complexity of a putative counterexample to the $p$-adic Littlewood conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Dmitry Badziahin, Dmitry Kleinbock, Manfred Einsiedler, Yann Bugeaud","submitted_at":"2014-05-21T20:16:13Z","abstract_excerpt":"Let $|| \\cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \\cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teuli\\'e asked whether $\\inf_{q \\ge 1} \\, q \\cdot || q \\alpha || \\cdot | q |_p = 0$ holds for every badly approximable real number $\\alpha$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $\\alpha$ grows too rapidly or too slowly, then their conjecture is true for the pair $(\\alpha, p)$ with $p$ an arbitrary prime."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5545","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":""},"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"}