On the superimposition of Christoffel words

TL;DR

This paper investigates the conditions under which two Christoffel words can be superimposed, providing necessary and sufficient criteria, counting possible superimpositions, and exploring new properties and geometric proofs related to Christoffel words.

Contribution

It establishes a complete characterization of superimposable Christoffel words, including counting superimpositions and deriving new properties and geometric insights.

Findings

Necessary and sufficient condition for superimposing two Christoffel words.

Number of possible superimpositions for given Christoffel words.

A new geometric proof of a classical money problem result.

Abstract

Initially stated in terms of Beatty sequences, the Fraenkel conjecture can be reformulated as follows: for a $k$-letter alphabet A, with a fixed $k \geq 3$, there exists a unique balanced infinite word, up to letter permutations and shifts, that has mutually distinct letter frequencies. Motivated by the Fraenkel conjecture, we study in this paper whether two Christoffel words can be superimposed. Following from previous works on this conjecture using Beatty sequences, we give a necessary and sufficient condition for the superimposition of two Christoffel words having same length, and more generally, of two arbitrary Christoffel words. Moreover, for any two superimposable Christoffel words, we give the number of different possible superimpositions and we prove that there exists a superimposition that works for any two superimposable Christoffel words. Finally, some new properties of Christoffel words are obtained as well as a geometric proof of a classic result concerning the money problem, using Christoffel words.