Normality and Finite-state Dimension of Liouville numbers

TL;DR

This paper presents new combinatorial and number-theoretic constructions of Liouville numbers that are normal in specific bases and have prescribed finite-state dimensions, refining the understanding of their complexity and distribution.

Contribution

It introduces simple, explicit constructions of Liouville numbers with any given finite-state dimension and normality properties, using combinatorial sequences and number-theoretic properties.

Findings

Constructed Liouville numbers normal in a given base using de Bruijn sequences.

Extended results to Liouville numbers with arbitrary finite-state dimensions.

Provided a method to construct Liouville numbers normal in finitely many bases under a conjecture.

Abstract

Liouville numbers were the first class of real numbers which were proven to be transcendental. It is easy to construct non-normal Liouville numbers. Kano and Bugeaud have proved, using analytic techniques, that there are normal Liouville numbers. Here, for a given base k >= 2, we give two simple constructions of a Liouville number which is normal to the base k. The first construction is combinatorial, and is based on de Bruijn sequences. A real number in the unit interval is normal if and only if its finite-state dimension is 1. We generalize our construction to prove that for any rational r in the closed unit interval, there is a Liouville number with finite state dimension r. This refines Staiger's result that the set of Liouville numbers has constructive Hausdorff dimension zero, showing a new quantitative classification of Liouville numbers can be attained using finite-state dimension. In the second number-theoretic construction, we use an arithmetic property of numbers - the existence of primitive roots - to construct Liouville numbers normal in finitely many bases, assuming a Generalized Artin's conjecture on primitive roots.