Cantor Series Constructions Contrasting Two Notions of Normality
Abstract
A. R\'enyi \cite{Renyi} made a definition that gives a generalization of simple normality in the context of -Cantor series. In \cite{Mance}, a definition of -normality was given that generalizes the notion of normality in the context of -Cantor series. In this work, we examine both -normality and -distribution normality, treated in \cite{Laffer} and \cite{Salat}. Specifically, while the non-equivalence of these two notions is implicit in \cite{Laffer}, in this paper, we give an explicit construction witnessing the nontrivial direction. That is, we construct a base as well as a real that is -normal yet not -distribution normal. We next approach the topic of simultaneous normality by constructing an explicit example of a base as well as a real that is both -normal and -distribution normal.
Cite
@article{arxiv.0911.4277,
title = {Cantor Series Constructions Contrasting Two Notions of Normality},
author = {Christian Altomare and Bill Mance},
journal= {arXiv preprint arXiv:0911.4277},
year = {2011}
}