Approximating Bernoulli words of irrational numbers by $\alpha$-words

Abstract

The Bernoulli word $B\left(\xi \right)$ of an irrational number $\xi$ is an infinite word over the alphabet $\{a,b\}$, in which the $n$th letter is $a$ if $[(n+1)\xi +\frac{1}{2}]-[n\xi +\frac{1}{2}]=0$ and is $b$ if $[(n+1)\xi +\frac{1}{2}]-[n\xi +\frac{1}{2}]=1$ $(n\ge 1)$. It is known that both $B\left(\xi \right)$ and the characteristic word $C\left(\xi \right)$ of $\xi$ are Sturmian words, and $C\left(\xi \right)$ is the limit of a sequence of standard words corresponding to $\xi$. In this paper, we determine a sequence of $\alpha$-words corresponding to $\xi$ which converges to $B\left(\xi \right)$. Our results are mainly based on the continued fraction expansion of $\xi$ and a result appears in a book of Venkov (1970).