  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 Teuli\'e 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.
