The spectral theory of regular sequences

TL;DR

This paper develops a measure-theoretic framework to analyze the asymptotic behavior of regular sequences, generalizing concepts from harmonic analysis and iterated function systems to understand their limiting properties.

Contribution

It introduces a systematic measure-theoretic approach to study the asymptotics of regular sequences, extending harmonic analysis methods to this class.

Findings

Constructed measures generalize mass distributions on attractors of iterated function systems.

Provided a new framework for understanding the limiting behavior of regular sequences.

Linked harmonic analysis of substitution measures to the asymptotics of regular sequences.

Abstract

Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Using the harmonic analysis of measures associated with substitutions as motivation, we study the limiting asymptotics of regular sequences by constructing a systematic measure-theoretic framework surrounding them. The constructed measures are generalisations of mass distributions supported on attractors of iterated function systems.