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

Let $\mathrm{\mid }\mathrm{\mid }\cdot \mathrm{\mid }\mathrm{\mid }$ denote the distance to the nearest integer and, for a prime number $p$, let $\mathrm{\mid }\cdot {\mathrm{\mid }}_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teuli'e asked whether $\inf_{q \ge 1} , q \cdot || q \alpha || \cdot | q |_p = 0$holdsforeverybadlyapproximablerealnumber$\alpha$ 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.