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.
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Notes
A proof of this result can be found in [15].
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Acknowledgements
We would like to thank Ted Slaman for many helpful discussions, and for first observing the relation between the granularity function of a measure and the settling time of a real. This crucial initial insight inspired much of the work presented here.
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Li, M., Reimann, J. Turing degrees and randomness for continuous measures. Arch. Math. Logic 63, 39–59 (2024). https://doi.org/10.1007/s00153-023-00873-7
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DOI: https://doi.org/10.1007/s00153-023-00873-7
Keywords
- Algorithmic randomness
- Continuous measures
- Turing degrees
- Recursively enumerable and above
- Moduli of computation