Normal and pseudonormal numbers

TL;DR

This paper explores the concept of normal and pseudonormal numbers, introducing probabilistic characterizations and a finite-step test for pseudonormality, advancing understanding of number normality in real analysis.

Contribution

It introduces the notion of pseudonormality, provides probabilistic characterizations of normality, and proposes a finite-step method to evaluate pseudonormality.

Findings

Normal numbers characterized by digit independence

Pseudonormality condition is verifiable in finite steps

Borel numbers linked to probability functions on base representations

Abstract

After a short review of the historical milestones on normal numbers, we introduce the Borel numbers as the reals admitting a probability function on their different bases representations. In this setting, we provide two probabilistic characterizations of normality based on the stochastic independence of their digits. Finally, we define the pseudonormality condition, which is satisfied by normal numbers and may be evaluated in a finite number of steps.