The Ghost Measures of Affine Regular Sequences

TL;DR

This paper explicitly calculates the ghost measure for affine 2-regular sequences, revealing their self-similar structure and measure properties, advancing understanding of these sequences' limiting behavior.

Contribution

It provides a detailed computation of the ghost measure and its Lebesgue decomposition for affine 2-regular sequences, a novel analysis in this area.

Findings

Explicit form of the ghost measure for affine 2-regular sequences

Description of the measure's Lebesgue decomposition

Insights into the self-similar structure of these sequences

Abstract

The k-regular sequences exhibit a self-similar behaviour between powers of k. One way to study this self-similarity is to attempt to describe the limiting shape of the sequence using measures, which results in an object which we call the ghost measure. The aim of this paper is to explicitly calculate this measure and some of its properties, including its Lebesgue decomposition, for the general family of affine 2-regular sequences.