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

D. Badziahin

Abstract

The $p$-adic Littlewood conjecture (PLC) states that $\underset{q\to \mathrm{\infty }}{lim inf}q\cdot |q{|}_p\cdot ||qx||=0$ 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 show that limit elements of the sequence $\{{\mathrm{T}}^n{w}_{CF}(x){\}}_{n\in \mathbb{N}}$ are quite natural objects to investigate in attempt to attack PLC for $x$. We then get several quite restrictive conditions on such limit elements $w$. As a consequence we prove that we must have $\underset{n\to \mathrm{\infty }}{lim}P(w,n)-n=\mathrm{\infty }$ 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.

Keywords: $p$-adic Littlewood conjecture, word complexity, continued fractions.