We give an algorithm to compute an absolutely normal number so that the first n digits in its binary expansion are obtained in time polynomial in n; in fact, just above quadratic. The algorithm uses combinatorial tools to ...

Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0,1,…,s-1 occurs with the same frequency 1/s. Let S be the set of positive integers that are ...

We demonstrate the full logical independence of normality to multiplicatively independent bases. This establishes that the set of bases to which a real number can be normal is not tied to any arithmetical properties other ...

We give an algorithm that computes an absolutely normal Liouville number.

We show that the set of absolutely normal numbers is Π03-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π03-complete in the effective Borel hierarchy.

We give a construction of an absolutely normal real number x such that for every integer b greater than or equal to 2, the discrepancy of the first N terms of the sequence (bnx mod 1)n≥0 is of asymptotic order O(N−1/2). ...

We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the firstn digits of its continued fraction expansion performs in the order ofn^4 mathematical ...

We give an elementary and direct proof of the following theorem: A real number is normal to a given integer base if, and only if, its expansion in that base is incompressible by lossless finite-state compressors (these are ...