Some Bounds Related to the 2-adic Littlewood Conjecture

2025/06/04 by Dinis Vitorino, Vitorino, Dinis, Ingrid Vukusic +1 · 1 citation
Mathematics · #10A30 #11A55 #11J13 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.04110

openalex publication_date 2025/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For every irrational real α, let M(α) = sup_{n≥ 1} a_n(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(2^k 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) = \limsup_{n→ ∞} a_n(α). In this setting, we prove that if B(2^k 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, …, 2^K β all equivalent to α.

Citations

• Estimating lower limit in the p-adic Littlewood conjecture
• A Note on the Base-p Expansions of Putative Counterexamples to the p-adic Littlewood Conjecture
• On the evolution of continued fractions in a fixed quadratic field
• Measure rigidity and p-adic Littlewood-type problems
• Quadratic irrational integers with partly prescribed continued fraction expansion

Cited by

• Estimating lower limit in the p-adic Littlewood conjecture

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