On a family of singular continuous measures related to the doubling map
arXiv:2006.09755 · doi:10.1016/j.indag.2021.06.001
Abstract
Here, we study some measures that can be represented by infinite Riesz products of 1-periodic functions and are related to the doubling map. We show that these measures are purely singular continuous with respect to Lebesgue measure and that their distribution functions satisfy super-polynomial asymptotics near the origin, thus providing a family of extremal examples of singular measures, including the Thue--Morse measure.
15 pages, with illustrating examples and figures, revised and slightly expanded version