Some complexity results in the theory of normal numbers

Abstract

Let $\mathcal{N}(b)$ be the set of real numbers which are normal to base $b$. A well-known result of H. Ki and T. Linton is that $\mathcal{N}(b)$ is ${\mathbf{\Pi }}_3^0$-complete. We show that the set $\mathcal{N}(b)$ of reals which preserve $\mathcal{N}(b)$ under addition is also ${\mathbf{\Pi }}_3^0$-complete. We use the characteriztion of $\mathcal{N}(b)$ given by G. Rauzy in terms of an entropy-like quantity called the noise. It follows from our results that no further characteriztion theorems could result in a still better bound on the complexity of $\mathcal{N}(b)$. We compute the exact descriptive complexity of other naturally occurring sets associated with noise. One of these is complete at the ${\mathbf{\Pi }}_4^0$ level. Finally, we get upper and lower bounds on the Hausdorff dimension of the level sets associated with the noise.

A talk explaining this paper may be found at https://www.youtube.com/watch?v=epgqcSkif2U&t=374s

References in corpus (1)

• On the Normality of Numbers to Different Bases

Cited by in corpus (1)

• Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property