The sixth lecture on algorithmic randomness

Abstract

This paper follows on from the author's Five Lectures on Algorithmic Randomness. It is concerned with material not found in that long paper, concentrating on Martin-Löf lowness and triviality. We present a hopefully user-friendly account of the decanter method, and discuss recent results of the author with Peter Cholak and Noam Greenberg concerning the class of strongly jump traceable reals introduced by Figueira, Nies and Stephan.

Beyond strong jump traceability

“…set that is K-trivial is jump traceable at identity? An analysis of the Decanter method in Downey [5] shows that if A is K-trivial, then A is jump traceable at all orders h such that ∞ n=1 h(n) −1 < ∞ (see [1,Theorem 1.3]). We do not know if our method adapts to the case h(n) = n. We also mention that Barmpalias, Downey and Greenberg [1] have refuted Conjecture 2.1 by a direct construction.…”

“…This works in the case when h is the identity order function (otherwise we just make suitable adjustments). For more details on this, see [5,Theorem 7.3]. Now we describe the positive requirements.…”