Construction of normal numbers with respect to the $Q$-Cantor series expansion for certain $Q$

TL;DR

This paper generalizes the concept of normality within the framework of $Q$-Cantor series, providing a method to construct $Q$-normal numbers by concatenating digit sequences with specific properties.

Contribution

It introduces a new definition of $Q$-normality and proves a theorem enabling the construction of $Q$-normal numbers through sequence concatenation.

Findings

Established a new $Q$-normality definition.

Proved a theorem for constructing $Q$-normal numbers.

Constructed explicit examples of $Q$-normal numbers.

Abstract

A. Renyi \cite{Renyi} made a definition that gives one generalization of simple normality in the context of $Q$-Cantor series. Similarly, in this paper we give a definition which generalizes the notion of normality in the context of $Q$-Cantor series. We will prove a theorem that allows us to concatenate sequences of digits that have a special property to give us the digits of a $Q$-normal number for certain $Q$. We will then use this theorem to construct a Q and a real number $x$ that is $Q$-normal.