On continued fraction expansion of potential counterexamples to $p$-adic Littlewood conjecture

Abstract

The $p$-adic Littlewood conjecture (PLC) states that $\liminf_{q\to\infty} q\cdot |q|_p \cdot ||qx|| = 0$ for every prime $p$ and every real $x$. Let $w_{CF}(x)$ be an infinite word composed of the continued fraction expansion of $x$ and let $\mathrm{T}$ be the standard left shift map. Assuming that $x$ is a counterexample to PLC we get several restrictions on limit elements of the sequence $\{\mathrm{T}^n w_{CF}(x)\}_{n\in\mathbb{N}}$. As a consequence we show that for any such limit element $w$ we must have $\lim_{n\to\infty} P(w,n) - n = \infty$ where $P(w,n)$ is a word complexity of $w$. We also show that $w$ can not be among a certain collection of recursively constructed words.