A note on the base-p expansions of putative counterexamples to the p-adic Littlewood conjecture

In this paper, we investigate the base-$p$ expansions of putative counterexamples to the $p$-adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base-$p$ expansion is uniformly recurrent. Furthermore, we show that if the base-$p$ expansion of $x$ is a morphic word $\tau \left({\phi }^{\omega }\left(a\right)\right)$ where ${\phi }^{\omega }\left(a\right)$ contains a subword of the form $uXuXu$ with ${lim}_{n\to \infty }\left|{\phi }^n\left(u\right)\right|=\infty$, then $x$ satisfies the $p$-adic Littlewood conjecture. In the special case when $p=2$, we show that the conjecture holds for all pure morphic words.