2021/08/10 by Michael James Coons, James S. Evans, Coons, Michael +6
Computer Science · Mathematics · #11B85 #28A80 #Cellular Automata and Applications #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2108.05007
openalex publication_date 2021/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Ghost measures of regular sequences---the unbounded analogue of automatic\nsequences---are generalisations of standard fractal mass distributions. They\nwere introduced to determine fractal (or self-similar) properties of regular\nsequences similar to those related to automatic sequences. The existence and\ncontinuity of ghost measures for a large class of regular sequences was\nrecently given by Coons, Evans and Ma ~nibo. In this paper, we provide an\nexplicit connection between fractals and regular sequences by showing that the\ngraphs of ghost distributions---the distribution functions of ghost\nmeasures---of the above-mentioned class of regular sequences are sections of\nself-affine sets. As an application of our result, we show that the ghost\ndistributions of the Zaremba sequences---regular sequences of the denominators\nof the convergents of badly approximable numbers---are all singular continuous.\n