2019/10/24 by Mingyang Li, Li, Mingyang, Jan Reimann +1
Computer Science · Mathematics · #03D25 #03D28 #03D32 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1910.11213
openalex publication_date 2019/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Δ02-degree contains an NCR element.