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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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##### References:
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