## 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 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 $$\lim_{n\to\infty} P(w,n) - n = \infty$$ 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.

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

This paper is available as a pdf (256kB) file. It is also on the arXiv: arxiv.org/abs/1406.3594.

 Sunday, July 23, 2017