$α$-numbers, diophantine exponent and factorisations of sturmian words
Caius Wojcik
TL;DR
The paper develops a unified Ostrowski- and Rauzy-graph framework for Sturmian words by introducing $\alpha$-numbers and formal intercepts, proving that every Sturmian word of slope $\alpha$ is a shift of the characteristic word $c_\alpha$ by a unique $\alpha$-number $\rho$, and deriving explicit formulas for the repetition function and diophantine exponent in terms of continuants $q_n$ and Ostrowski digits. It then builds a robust factorisation theory, including equivalence of $\alpha$-numbers, infinite-product representations, and precise conditions for factorisations and suffix structures, culminating in torsion-type relations among $\alpha$-numbers and illustrating these phenomena with Fibonacci-like examples. The results connect symbolic dynamics, Diophantine approximation, and combinatorics on words, providing constructive descriptions of factorisations and revealing deep arithmetic structure behind Sturmian sequences. Overall, the work offers new tools to compute diophantine exponents, classify intercepts, and understand the factorisation landscape of Sturmian words via Ostrowski-based encodings.
Abstract
We introduce the notion of $α$-numbers and formal intercept of sturmian words, and derive from this study general factorisations formula for sturmian words. Sturmian words are defined as infinite words with lowest unbound complexity, and are characterized by two parameters, the first one being well-known as the slope, and the second being their formal intercepts. We build this formalism by a study of Rauzy graphs of sturmian words, and we use this caracterisation to compute the repetition function of sturmian words and their diophantine exponent. We then develop these techniques to provide general factorisations formulas for sturmian words.