**Title**

An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals

**Type**

Fullpaper

**Journal**

Notre Dame Journal of Formal Logic, 51(3):337-349, 2010.

Permanent link to this document in this Journal.

Received November 30, 2009; accepted December 8, 2009; printed June 16, 2010

**Download**

preprint

Japanese summary

**Abstract**

Van Lambalgen’s Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen’s Theorem by considerint the join of infinitely many reals which are random relative to each other.

In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi’}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$. We also prove that there exists A such that, for each n, the real $\Omega^A_M$ is highn for some universal Turing machine M by using the extended van Lambalgen’s Theorem.

**Cited by**

@article{bienvenu2010ergodic,

title={{Ergodic-type characterizations of algorithmic randomness}},

author={Bienvenu, L. and Day, A. and Mezhirov, I. and Shen, A.},

journal={Programs, Proofs, Processes},

pages={49–58},

year={2010},

publisher={Springer}

}

@article{bienvenu2011constructive,

title={A constructive version of Birkhoffʼs ergodic theorem for Martin–L{\”o}f random points},

author={Bienvenu, L. and Day, A. and Hoyrup, M. and Mezhirov, I. and Shen, A.},

journal={Information and Computation},

year={2011},

publisher={Elsevier}

}

@article{yu2011characterizing,

title={Characterizing strong randomness via Martin-L{\”o}f randomness},

author={Yu, L.},

journal={Annals of Pure and Applied Logic},

year={2011},

publisher={Elsevier}

}