Ghost distributions of regular sequences are affine transformations of self-affine sets
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.
Keywords
Cite
@article{arxiv.2108.05007,
title = {Ghost distributions of regular sequences are affine transformations of self-affine sets},
author = {Michael Coons and James Evans and Zachary Groth and Neil Mañibo},
journal= {arXiv preprint arXiv:2108.05007},
year = {2021}
}
Comments
15 pages