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

Abstract

The $p$-adic Littlewood conjecture (PLC) states that ${\mathrm{liminf}}_{q\to \infty }q\cdot |q{|}_p\cdot ||qx||=0$ 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.

20 pages

Cited by in corpus (1)

• Infinite Loops and the $p$-adic Littlewood Conjecture. Part I: Reformulating the $p$-adic Littlewood Conjecture in Terms of Infinite Loops