Higher Kurtz randomness

TL;DR

This paper explores a higher form of Kurtz randomness within the analytical hierarchy, establishing the existence of a cone of $oldsymbol{ ext{Pi}}^1_1$-Kurtz random hyperdegrees and characterizing lowness properties.

Contribution

It introduces the concept of higher Kurtz randomness and characterizes lowness for this randomness in terms of $oldsymbol{ ext{Delta}}^1_1$-dominated and semi-traceable degrees.

Findings

Existence of a cone of $oldsymbol{ ext{Pi}}^1_1$-Kurtz random hyperdegrees.

Characterization of lowness for $oldsymbol{ ext{Delta}}^1_1$-Kurtz randomness.

Connection between higher Kurtz randomness and descriptive set theory.

Abstract

A real $x$ is $\Delta^1_1$-Kurtz random ($\Pi^1_1$-Kurtz random) if it is in no closed null $\Delta^1_1$ set ($\Pi^1_1$ set). We show that there is a cone of $\Pi^1_1$-Kurtz random hyperdegrees. We characterize lowness for $\Delta^1_1$-Kurtz randomness as being $\Delta^1_1$-dominated and $\Delta^1_1$-semi-traceable.