Coarse computability, the density metric, Hausdorff distances between Turing degrees, perfect trees, and reverse mathematics

TL;DR

This paper explores the structure of coarse similarity classes and Turing degrees using a new metric based on density and Hausdorff distance, revealing their topological, computability, and reverse mathematical properties.

Contribution

It introduces a novel metric space for coarse similarity classes and Turing degrees, analyzing their geometric, measure-theoretic, and logical characteristics.

Findings

Between any two points, there are continuum many geodesic paths.

Distances between Turing degrees are exactly 0, 1/2, or 1.

Distribution of attractive and dispersive degrees is studied.

Abstract

The coarse similarity class $[A]$ of $A$ is the set of all $B$ whose symmetric difference with $A$ has asymptotic density 0. There is a natural metric $\delta$ on the space $\mathcal{S}$ of coarse similarity classes defined by letting $\delta([A],[B])$ be the upper density of the symmetric difference of $A$ and $B$. We study the resulting metric space, showing in particular that between any two distinct points there are continuum many geodesic paths. We also study subspaces of the form $\{[A] : A \in \mathcal U\}$ where $\mathcal U$ is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of $\mathcal U$. We then define a distance between Turing degrees based on Hausdorff distance in this metric space. We adapt a proof of Monin to show that the distances between degrees that occur are exactly 0, 1/2, and 1, and study which of these values occur most frequently in the senses of measure and category. We define a degree to be attractive if the class of all degrees at distance 1/2 from it has measure 1, and dispersive otherwise. We study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition. We also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic. Finally, we study the completeness of $(\mathcal S,\delta)$ from the perspectives of computability theory and reverse mathematics.