Some Bounds Related to the $2$-adic Littlewood Conjecture
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913986117632000 |
|---|---|
| author | Vitorino, Dinis Vukusic, Ingrid |
| author_facet | Vitorino, Dinis Vukusic, Ingrid |
| contents | For every irrational real $α$, let $M(α) = \sup_{n\geq 1} a_n(α)$ 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 $α$ such that $M(2^k α)$ 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 $M(α)$ by $B(α) = \limsup_{n\to \infty} a_n(α)$. In this setting, we prove that if $B(2^k α) \leq C$ 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 $α$ with the property that for arbitrarily large $K$ there exist $β, 2β, 4 β, \ldots, 2^K β$ all equivalent to $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04110 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Bounds Related to the $2$-adic Littlewood Conjecture Vitorino, Dinis Vukusic, Ingrid Number Theory 11A55, 10A30, 11J13 For every irrational real $α$, let $M(α) = \sup_{n\geq 1} a_n(α)$ 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 $α$ such that $M(2^k α)$ 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 $M(α)$ by $B(α) = \limsup_{n\to \infty} a_n(α)$. In this setting, we prove that if $B(2^k α) \leq C$ 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 $α$ with the property that for arbitrarily large $K$ there exist $β, 2β, 4 β, \ldots, 2^K β$ all equivalent to $α$. |
| title | Some Bounds Related to the $2$-adic Littlewood Conjecture |
| topic | Number Theory 11A55, 10A30, 11J13 |
| url | https://arxiv.org/abs/2506.04110 |