Normal equivalencies for eventually periodic basic sequences

TL;DR

This paper investigates when normality with respect to an eventually periodic basic sequence is equivalent to normality in a certain base, extending previous work on normality equivalences and showing boundedness alone is insufficient.

Contribution

It establishes the conditions under which $Q$-normality and $Q$-distribution normality are equivalent to base-$b$ normality for eventually periodic basic sequences.

Findings

$Q$-normality is equivalent to base-$b$ normality for certain $Q$

Boundedness of $Q$ alone does not guarantee equivalence

Equivalence depends on the specific structure of $Q$

Abstract

W. M. Schmidt, A. D. Pollington, and F. Schweiger have studied when normality with respect to one expansion is equivalent to normality with respect to another expansion. Following in their footsteps, we show that when $Q$ is an eventually periodic basic sequence, that $Q$-normality and $Q$-distribution normality are equivalent to normality in base $b$ where $b$ is dependent on $Q$. We also show that boundedness of the basic sequence is not sufficient for this equivalence.