Cantor series constructions of sets of normal numbers

TL;DR

This paper constructs sets of $Q$-distribution normal numbers using Cantor series, providing explicit examples, Hausdorff dimension results, and discrepancy estimates under certain conditions.

Contribution

It offers a novel construction of $Q$-distribution normal numbers for arbitrary sequences $Q$, including explicit examples and dimension estimates.

Findings

Constructed perfect, nowhere dense sets of $Q$-distribution normal numbers.

Provided explicit examples with prescribed Hausdorff dimension.

Established discrepancy estimates under growth conditions on $q_n$.

Abstract

Let $Q=(q_n)_{n=1}^{\infty}$ be a sequence of integers greater than or equal to 2. We say that a real number $x$ in $[0,1)$ is {\it $Q$-distribution normal} if the sequence $(q_1q_2... q_n x)_{n=1}^{\infty}$ is uniformly distributed mod 1. In \cite{Lafer}, P. Lafer asked for a construction of a $Q$-distribution normal number for an arbitrary $Q$. Under a mild condition on $Q$, we construct a set $\Theta_Q$ of $Q$-distribution normal numbers. This set is perfect and nowhere dense. Additionally, given any $\alpha$ in $[0,1]$, we provide an explicit example of a sequence $Q$ such that the Hausdorff dimension of $\Theta_Q$ is equal to $\alpha$. Under a certain growth condition on $q_n$, we provide a discrepancy estimate that holds for every $x$ in $\Theta_Q$.