PUB - Publikationen an der Universität Bielefeld

Motivated by near-identical graphs of two increasing continuous functions-one related to Zaremba's conjecture and the other due to Salem-we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of self-affine sets. Additionally, we provide a sufficient condition for such measures to be purely singular continuous. As a corollary, and analogous to Salem's strictly increasing singular continuous function, we show that the ghost distributions of the Zaremba sequences are singular continuous.