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

TL;DR

This paper investigates the structure of potential counterexamples to the $p$-adic Littlewood conjecture by analyzing their continued fraction expansions and showing restrictions on their limit behaviors and complexity.

Contribution

It introduces new restrictions on the continued fraction expansions of counterexamples to the $p$-adic Littlewood conjecture, including complexity growth and exclusion from certain recursive word classes.

Findings

Limit elements of the shift orbit have unbounded complexity growth.

Counterexamples cannot be among certain recursively constructed words.

Provides structural constraints on potential counterexamples.

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.