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Energy randomness. (English) Zbl 06976569
Summary: 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 \(X \in 2^\omega\) is \(s\)-energy random if and only if \(\sum_{n \in \omega} 2^{sn - KM(X \upharpoonright n)} < \infty\), providing a characterization of energy randomness via a priori complexity \(KM\). This is related to a question of Allen, Bienvenu, and Slaman.

MSC:
03D32 Algorithmic randomness and dimension
31C15 Potentials and capacities on other spaces
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