{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/some-bounds-related-to-the-2-adic-littlewood","title":"Some Bounds Related to the 2-adic Littlewood Conjecture","arxiv_id":"2506.04110","date":"2025-06-04","proceeding":null,"authors":["Dinis Vitorino","Ingrid Vukusic"],"abstract":"For every irrational real $\\alpha$, let $M(\\alpha) = \\sup_{n\\geq 1} a_n(\\alpha)$ denote the largest partial quotient in its continued fraction expansion (or $\\infty$, if unbounded). The $2$-adic Littlewood conjecture (2LC) can be stated as follows: There exists no irrational $\\alpha$ such that $M(2^k \\alpha)$ is uniformly bounded by a constant $C$ for all $k\\geq 0$. In 2016, Badziahin proved (considering a different formulation of 2LC) that if a counterexample exists, then the bound $C$ is at least $8$. We improve this bound to $15$. Then we focus on a ``B-variant'' of 2LC, where we replace $M(\\alpha)$ by $B(\\alpha) = \\limsup_{n\\to \\infty} a_n(\\alpha)$. In this setting, we prove that if $B(2^k \\alpha) \\leq C$ for all $k\\geq 0$, then $C \\geq 5$. For the proof we use Hurwitz's algorithm for multiplication of continued fractions by 2. Along the way, we find families of quadratic irrationals $\\alpha$ with the property that for arbitrarily large $K$ there exist $\\beta, 2\\beta, 4 \\beta, \\ldots, 2^K \\beta$ all equivalent to $\\alpha$.","url_abs":"https://arxiv.org/abs/2506.04110v1","url_pdf":"https://arxiv.org/pdf/2506.04110v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"some-bounds-related-to-the-2-adic-littlewood","repo_url":"https://github.com/dinisvit/computer_aided_2LC","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}