On the complexity of a putative counterexample to the $p$-adic Littlewood conjecture

Abstract

Let $||\cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $|\cdot {|}_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teulié asked whether ${\inf }_{q\ge 1}q\cdot ||q\alpha ||\cdot |q{|}_p=0$ holds for every badly approximable real number $\alpha$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $\alpha$ grows too rapidly or too slowly, then their conjecture is true for the pair $(\alpha ,p)$ with $p$ an arbitrary prime.

19 pages; a revised version