Rhythmic generation of infinite trees and languages

TL;DR

This paper explores the properties of periodic signatures in infinite trees and languages, linking them to rational base numeration systems and characterizing languages with purely periodic signatures as representations of integers in these systems.

Contribution

It introduces a detailed analysis of periodic signatures in infinite trees and languages, connecting them to rational base numeration systems and characterizing their structure.

Findings

Periodic signatures relate to rational base numeration systems.

Languages with purely periodic signatures represent integers in non-canonical digit systems.

The work characterizes when signatures are purely periodic based on number system representations.

Abstract

This work builds on the notion of breadth-first signature of infinite trees and (prefix-closed) languages introduced by the authors in a previous work. We focus here on periodic signatures, a case coming from the study of rational base numeration systems; the language of integer representations in base~$\frac{p}{q}$ has a purely periodic signature whose period is derived from the Christoffel word of slope~$\frac{p}{q}$. Conversely, we characterise languages whose signature are purely periodic as representations of integers in such number systems with non-canonical alphabets of digits.