Papers › Some Bounds Related to the 2-adic Littlewood Conjecture
Some Bounds Related to the 2-adic Littlewood Conjecture
Dinis Vitorino, Ingrid Vukusic
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
For every irrational real α, let M(α) = sup_(n≥1) aₙ(α) denote the largest partial quotient in its continued fraction expansion (or ∞, if unbounded). The $2$-adic Littlewood conjecture (2LC) can be stated as follows: There exists no irrational α such that M(2ᵏ α) is uniformly bounded by a constant C for all k≥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(α) by B(α) = lim sup_(n→∞) aₙ(α). In this setting, we prove that if B(2ᵏ α) ≤C for all k≥0, then C ≥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 α with the property that for arbitrarily large K there exist β, 2β, 4 β, …, 2ᴷ β all equivalent to α.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections