A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture

Abstract

In this paper, we investigate the base-$p$ expansions of putative counterexamples to the $p$-adic Littlewood conjecture of de Mathan and Teuli\'e. 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 $\tau(\phi^\omega(a))$ where $\phi^\omega(a)$ contains a subword of the form $uXuXu$ with $\lim_{n\to\infty}|\phi^n(u)|=\infty$, 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.