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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.04110","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-06-04T16:01:54Z","cross_cats_sorted":[],"title_canon_sha256":"3ad426e768409369f3682e8ed866577936a144b22ec564085ec2979eb9d83c84","abstract_canon_sha256":"bfa218d32bfa9708199fb13b8aa4565d2415a8f6a9b17198660cb33c71fc093e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:52:20.924646Z","signature_b64":"aX0dzAZQJq6+quz0Hiy+c0QYt2dQi/L14/+/RQ72NFd+vAGrC6Jgwl+0k/WqD5Y7TqFRbdDGiyWDZE/ku+dlBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50960bd5df76cc32747666b19570d4bfe20f62560662c55f3dce52f24fb93444","last_reissued_at":"2026-07-05T11:52:20.924096Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:52:20.924096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Bounds Related to the $2$-adic Littlewood Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dinis Vitorino, Ingrid Vukusic","submitted_at":"2025-06-04T16:01:54Z","abstract_excerpt":"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). 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