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A Universal Separable Diversity

David Bryant;  André Nies;  Paul Tupper — 2017

Analysis and Geometry in Metric Spaces

The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant...

Coherent randomness tests and computing the K -trivial sets

Laurent Bienvenu;  Noam Greenberg;  Antonín Kučera;  André Nies;  Dan Turetsky — 2016

Journal of the European Mathematical Society

We introduce Oberwolfach randomness, a notion within Demuth’s framework of statistical tests with moving components; here the components’ movement has to be coherent across levels. We show that a ML-random set computes all K -trivial sets if and only if it is not Oberwolfach random, and indeed that there is a K -trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis, such as the...

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