Kolakoski-(2m,2n) are limit-periodic model sets

TL;DR

This paper proves that generalized Kolakoski sequences with two even numbers are limit-periodic model sets with pure point spectra, linking substitution rules, dynamical systems, and internal space visualization.

Contribution

It establishes the pure point dynamical and diffractive spectra of these sequences and connects them to model sets with ell-adic internal spaces.

Findings

Sequences have pure point spectrum

Sequences are limit-periodic model sets

Internal spaces can be visualized

Abstract

We consider (generalized) Kolakoski sequences on an alphabet with two even numbers. They can be related to a primitive substitution rule of constant length ell. Using this connection, we prove that they have pure point dynamical and pure point diffractive spectrum, where we make use of the strong interplay between these two concepts. Since these sequences can then be described as model sets with ell-adic internal space, we add an approach to ``visualize'' such internal spaces.