On the Hausdorff dimension of some sets of numbers defined through the digits of their $Q$-Cantor series expansions

Dylan Airey, Bill Mance

Published 2014-07-03, updated 2014-07-15Version 2

Following in the footsteps of P. Erd\H{o}s and A. R\'enyi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we consider difference sets associated with various notions of normality and sets of numbers with a prescribed range of digits.