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

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 $\alpha$.