Combinatorial structure of Sturmian words and continued fraction expansions of Sturmian numbers

TL;DR

This paper explores the combinatorial structure of Sturmian words and their connection to continued fraction expansions, extending classical results to non-characteristic Sturmian words and deriving formulas for irrationality exponents.

Contribution

It generalizes the structure of Sturmian words beyond characteristic cases and links their properties to continued fractions and irrationality measures.

Findings

Extended the combinatorial structure to all Sturmian words.

Derived continued fraction expansions for numbers with Sturmian base-b expansions.

Provided formulas for irrationality exponents based on Sturmian word parameters.

Abstract

Let $\theta = [0; a_1, a_2, \dots]$ be the continued fraction expansion of an irrational real number $\theta \in (0, 1)$. It is well-known that the characteristic Sturmian word of slope $\theta$ is the limit of a sequence of finite words $(M_k)_{k \ge 0}$, with $M_k$ of length $q_k$ (the denominator of the $k$-th convergent to $\theta$) being a suitable concatenation of $a_k$ copies of $M_{k-1}$ and one copy of $M_{k-2}$. Our first result extends this to any Sturmian word. Let $b \ge 2$ be an integer. Our second result gives the continued fraction expansion of any real number $\xi$ whose $b$-ary expansion is a Sturmian word ${\bf s}$ over the alphabet $\{0, b-1\}$. This extends a classical result of B\"ohmer who considered only the case where ${\bf s}$ is characteristic. As a consequence, we obtain a formula for the irrationality exponent of $\xi$ in terms of the slope and the intercept of ${\bf s}$.