A note on the base-[formula omitted] expansions of putative counterexamples to the [formula omitted]-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 τ(φω(a)) where φω(a) contains a subword of the form uXuXu with limn→∞|φn(u)|=∞, 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.