Normality and Finite-state Dimension of Liouville numbers (1204.4104v2)

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.