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arXiv · 2108.05007

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

Michael Coons·James Evans·Zachary Groth·Neil Mañibo
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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.

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Michael Coons, James Evans, Zachary Groth, Neil Mañibo. 2021-08-11. Ghost distributions of regular sequences are affine transformations of self-affine sets. https://arxiv.org/abs/2108.05007

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