Energy randomness

TL;DR

This paper characterizes energy randomness, a form of partial randomness, using a sum involving a priori complexity, linking it to the behavior of Martin-Löf random sequences and Brownian motion zero times.

Contribution

It provides a new characterization of energy randomness through a summation involving a priori complexity, answering a question posed by Allen, Bienvenu, and Slaman.

Findings

Energy randomness characterized by a convergent sum involving a priori complexity.

Established equivalence between energy randomness and a specific complexity sum.

Connected energy randomness to properties of Martin-Löf random sequences and Brownian motion.

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.