Some Bounds Related to the $2$-adic Littlewood Conjecture

TL;DR

This paper improves bounds related to the 2-adic Littlewood conjecture, showing that if certain boundedness conditions hold, the bounds must be at least 15 or 5, and explores properties of quadratic irrationals under these conditions.

Contribution

The paper refines existing bounds on the 2-adic Littlewood conjecture and introduces a B-variant, providing new lower bounds and analyzing quadratic irrationals with specific equivalence properties.

Findings

Bound C for 2LC improved to 15 from 8.

Bound C in B-variant improved to 5.

Existence of quadratic irrationals with specific equivalence properties.

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 $\alpha$.