A characterization of fine words over a finite alphabet

TL;DR

This paper characterizes 'fine' infinite words over finite alphabets, showing they are either strict episturmian or skew episturmian, generalizing previous results on Sturmian words.

Contribution

It provides a complete characterization of fine words over finite alphabets, extending known results from binary to arbitrary finite alphabets.

Findings

Fine words are exactly strict episturmian or skew episturmian words.

The characterization generalizes Pirillo's binary alphabet results.

The paper links lexicographic minimality to episturmian structures.

Abstract

To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is "fine" if there exists an infinite word u such that, for any lexicographic order, min(w) = au where a = min(A). In this paper, we characterize fine words; specifically, we prove that an infinite word w is fine if and only if w is either a "strict episturmian word" or a strict "skew episturmian word''. This characterization generalizes a recent result of G. Pirillo, who proved that a fine word over a 2-letter alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but not periodic) infinite word, all of whose factors are (finite) Sturmian.