Energy randomness

Abstract

Energy randomness is a notion of partial randomness introduced by Diamondstone and Kjos-Hanssen to characterize the sequences that can be elements of a Martin-L\"of random closed set (in the sense of Barmpalias, Brodhead, Cenzer, Dashti, and Weber). It has also been applied by Allen, Bienvenu, and Slaman to the characterization of the possible zero times of a Martin-L\"of random Brownian motion. 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.