Turing Degrees and Randomness for Continuous Measures

TL;DR

This paper explores the properties of reals that are not random with respect to any continuous measure, introducing new measure-based techniques and proving the existence of such reals in various Turing degrees, including all Δ^0_2 degrees.

Contribution

It introduces generalized Hausdorff measures and constructions that preserve non-randomness, establishing the presence of NCR reals across a wide range of Turing degrees.

Findings

Every Δ^0_2 degree contains an NCR real.

New measure-theoretic tools for analyzing non-randomness.

Existence of NCR reals in multiple Turing degrees.

Abstract

We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the "dissipation" function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every $\Delta^0_2$-degree contains an NCR element.