Liouville Numbers, Borel Normality and Algorithmic Randomness

Abstract

We present a systematic comparison between Liouville, computable, Borel normal and Martin-Lof random numbers. The nine non-empty combinations, all small in measure or category, are illustrated with concrete examples. The sets of Liouville numbers and Martin-Lof random numbers are disjoint, thus showing that the irrationality exponent is not a measure of randomness. Finally, we construct the first computable set of correlations appearing in every Martin-Lof random number, but not in all numbers.