On Kurtz randomness

Abstract

Kurtz randomness is a notion of algorithmic randomness for real numbers. In particular a real α is called Kurtz random (or weakly random) iff it is contained in every computably enumerable set U of (Lebesgue) measure 1. We prove a number of characterizations of this notion, relating it to other notions of randomness such as the well-known notions of computable randomness, Martin-Löf randomness and Schnorr randomness. For the first time we give machine characterizations of Kurtz randomness. Whereas the Turing degree of every Martin-Löf random c.e. real is the complete degree, and the degrees of Schnorr random c.e. reals are all high, we show that Kurtz random c.e. reals occur in every non-zero c.e. degree. Additionally, we show that the sets that are low for Kurtz randomness are all hyperimmune and include those that are low for Schnorr randomness, characterized previously by Terwijn and Zambella.