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Energy randomness

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Abstract

Energy randomness is a notion of partial randomness introduced by Kjos- Hanssen to characterize the sequences that can be elements of a Martin–Löf random closed set (in the sense of Barmpalias, Brodhead, Cenzer, Dashti and Weber). It brings together ideas from potential theory and algorithmic randomness. In this paper, we show that X ∈ 2ω is s-energy random if and only if \(\sum\nolimits_{n \in \omega } {{2^{sn - KM(X \upharpoonright n)}}} < \infty \), providing a characterization of energy randomness via a priori complexity KM. This is related to a question of Allen, Bienvenu, and Slaman.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    Joseph S. Miller

  2. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Jason Rute

Authors
  1. Joseph S. Miller
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  2. Jason Rute
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Correspondence to Joseph S. Miller.

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Miller, J.S., Rute, J. Energy randomness. Isr. J. Math. 227, 1–26 (2018). https://doi.org/10.1007/s11856-018-1731-z

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  • DOI: https://doi.org/10.1007/s11856-018-1731-z

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