{"paper":{"title":"On continued fraction expansion of potential counterexamples to $p$-adic Littlewood conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dzmitry Badziahin","submitted_at":"2014-06-13T17:09:09Z","abstract_excerpt":"The $p$-adic Littlewood conjecture (PLC) states that $\\liminf_{q\\to\\infty} q\\cdot |q|_p \\cdot ||qx|| = 0$ for every prime $p$ and every real $x$. Let $w_{CF}(x)$ be an infinite word composed of the continued fraction expansion of $x$ and let $\\mathrm{T}$ be the standard left shift map. Assuming that $x$ is a counterexample to PLC we get several restrictions on limit elements of the sequence $\\{\\mathrm{T}^n w_{CF}(x)\\}_{n\\in\\mathbb{N}}$. As a consequence we show that for any such limit element $w$ we must have $\\lim_{n\\to\\infty} P(w,n) - n = \\infty$ where $P(w,n)$ is a word complexity of $w$. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.3594","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"}