Cantor Series Constructions Contrasting Two Notions of Normality

TL;DR

This paper explores different notions of normality in Cantor series expansions, explicitly constructing examples that distinguish and simultaneously satisfy these notions, advancing understanding of their relationship.

Contribution

It provides explicit constructions of bases and reals that distinguish and unify two notions of normality in $Q$-Cantor series, clarifying their relationship.

Findings

Constructed a base $Q$ and real $x$ that is $Q$-normal but not $Q$-distribution normal.

Constructed a base $Q$ and real $x$ that is both $Q$-normal and $Q$-distribution normal.

Clarified the non-equivalence and potential coexistence of these normality notions.

Abstract

A. R\'enyi \cite{Renyi} made a definition that gives a generalization of simple normality in the context of $Q$-Cantor series. In \cite{Mance}, a definition of $Q$-normality was given that generalizes the notion of normality in the context of $Q$-Cantor series. In this work, we examine both $Q$-normality and $Q$-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 $Q$ as well as a real $x$ that is $Q$-normal yet not $Q$-distribution normal. We next approach the topic of simultaneous normality by constructing an explicit example of a base $Q$ as well as a real $x$ that is both $Q$-normal and $Q$-distribution normal.