Mass problems and almost everywhere domination

✍ Scribed by Stephen G. Simpson

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 144 KB
• Volume: 53
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200710013

✦ Synopsis

Abstract

We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the sets of reals which are almost everywhere dominating and Martin‐Löf random, respectively. Let b~1~, b~2~, and b~3~ be the degrees of unsolvability of the mass problems associated with AED, MLR × AED, and MLR ∩ AED, respectively. Let 𝒫~w~ be the lattice of degrees of unsolvability of mass problems associated with nonempty Π^0^~1~ subsets of 2^ω^ . Let 1 and 0 be the top and bottom elements of 𝒫~w~ . We show that inf(b~1~, 1), inf(b~2~, 1), and inf(b~3~, 1) belong to 𝒫~w~ and 0 < inf(b~1~, 1) < inf(b~2~, 1) < inf(b~3~, 1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into 𝒫~w~ , we show that inf(b~1~, 1) and inf(b~3~, 1) but not inf(b~2~, 1) are comparable with some recursively enumerable Turing degrees other than 0 and 0′. In order to make this paper more self‐contained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)