## Mass problems and almost everywhere domination (2007)

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Venue: | Mathematical Logic Quarterly |

Citations: | 10 - 7 self |

### BibTeX

@INPROCEEDINGS{Simpson07massproblems,

author = {Stephen G. Simpson},

title = {Mass problems and almost everywhere domination},

booktitle = {Mathematical Logic Quarterly},

year = {2007},

pages = {53--483}

}

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### Abstract

We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and Martin-Löf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw 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 Pw. We show that inf(b1,1) and inf(b2,1) and inf(b3,1) belong to Pw and that 0 < inf(b1,1) < inf(b2,1) < inf(b3,1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, we show that inf(b1,1) and inf(b3,1) but not inf(b2,1) 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.