On a metric generalization of the $tt$-degrees and effective dimension theory

TL;DR

This paper introduces a metric generalization of $tt$-degrees for points in computable metric spaces, linking it to Borel isomorphism and exploring implications in effective topological and fractal dimension theories.

Contribution

It defines and characterizes the metric $tt$-degree concept, extending classical notions to computable metric spaces and connecting it with effective dimension theories.

Findings

Characterization of metric $tt$-degrees via Borel isomorphism

Extension of $tt$-reducibility to computable metric spaces

Insights into effective topological and fractal dimension relationships

Abstract

In this article, we study an analogue of $tt$-reducibility for points in computable metric spaces. We characterize the notion of the metric $tt$-degree in the context of first-level Borel isomorphism. Then, we study this concept from the perspectives of effective topological dimension theory and of effective fractal dimension theory.