Ghost distributions of regular sequences are affine transformations of self-affine sets

TL;DR

This paper reveals that ghost distributions of regular sequences are affine transformations of self-affine sets, linking fractal geometry with the properties of these sequences, and shows that ghost distributions of Zaremba sequences are singular continuous.

Contribution

It establishes an explicit connection between ghost distributions of regular sequences and self-affine fractals, extending understanding of their geometric structure.

Findings

Ghost distributions are sections of self-affine sets.

Ghost distributions of Zaremba sequences are singular continuous.

Provides a geometric interpretation of ghost measures.

Abstract

Ghost measures of regular sequences---the unbounded analogue of automatic sequences---are generalisations of standard fractal mass distributions. They were introduced to determine fractal (or self-similar) properties of regular sequences similar to those related to automatic sequences. The existence and continuity of ghost measures for a large class of regular sequences was recently given by Coons, Evans and Ma\~nibo. In this paper, we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions---the distribution functions of ghost measures---of the above-mentioned class of regular sequences are sections of self-affine sets. As an application of our result, we show that the ghost distributions of the Zaremba sequences---regular sequences of the denominators of the convergents of badly approximable numbers---are all singular continuous.