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 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 $\tau ({\varphi }^{\omega }(a))$ where ${\varphi }^{\omega }(a)$ contains a subword of the form $uXuXu$ with ${\lim }_{n\to \infty }|{\varphi }^n(u)|=\infty$, 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.

14 pages, no figures