A computable absolutely normal Liouville number

We give an algorithm that computes an absolutely normal Liouville number. 1. The main result The set of Liouville numbers is {x ∈ R \Q : ∀k ∈ N, ∃q ∈ N, q > 1 and ||qx|| < q−k} where ||x|| = min{|x−m| : m ∈ Z} is the distance of a real number x to the nearest integer and other notation is as usual. Liouville’s constant, ∑ k≥1 10 −k!, is the standard example of a Liouville number. Though uncountable, the set of Liouville numbers is small, in fact, it is null, both in Lebesgue measure and in Hausdorff dimension (see [6]). We say that a base is an integer s greater than or equal to 2. A real number x is normal to base s if the sequence (sx : j ≥ 0) is uniformly distributed in the unit interval modulo one. By Weyl’s Criterion [11], x is normal to base s if and only if certain harmonic sums associated with (sx : j ≥ 0) grow slowly. Absolute normality is normality to every base. Bugeaud [6] established the existence of absolutely normal Liouville numbers by means of an almost-all argument for an appropriate measure due to Bluhm [3, 4]. The support of this measure is a perfect set, which we call Bluhm’s fractal, all of whose irrational elements are Liouville numbers. The Fourier transform of this measure decays quickly enough to ensure that those harmonic sums grow slowly on a set of measure one. Thus, Bugeaud’s proof exhibits a nonempty set but does not provide a construction of an absolutely normal Liouville number. A real number x is computable if there is a base s and an algorithm to output the digits for the base-s expansion of x, one after the other. In this note we show the following: Theorem. There is a computable absolutely normal Liouville number. Date: January 29, 2014 and, in revised form, April 14, 2014. 2000 Mathematics Subject Classification. Primary 11K16, 68-04; Secondary 11-04.