Some bounds related to the 2‐adic Littlewood conjecture

For every irrational real , let 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 such that is uniformly bounded by a constant for all . In 2016, Badziahin proved (considering a different formulation of 2LC) that if a counterexample exists, then the bound is at least 8. We improve this bound to 15. Then we focus on a “B‐variant” of 2LC, where we replace by . In this setting, we prove that if for all , then . For the proof we use Hurwitz's algorithm for multiplication of continued fractions by 2. Along the way, we find families of quadratic irrationals with the property that for arbitrarily large there exist all equivalent to .