Degrees of and lowness for isometric isomorphism

TL;DR

This paper extends computable structure theory to metric structures, showing that lowness for isomorphism, isometric isomorphism, and isometry coincide, and classifies degrees low for isometric isomorphism in specific Lebesgue spaces.

Contribution

It establishes the equivalence of lowness notions for isomorphism and isometry in metric structures and classifies low degrees for isometric isomorphism in Lebesgue spaces.

Findings

Lowness for isomorphism coincides with lowness for isometric isomorphism.

Classifications of degrees low for isometric isomorphism in Lebesgue spaces.

Identifies conditions under which lowness notions are equivalent.

Abstract

We contribute to the program of extending computable structure theory to the realm of metric structures by investigating lowness for isometric isomorphism of metric structures. We show that lowness for isomorphism coincides with lowness for isometric isomorphism and with lowness for isometry of metric spaces. We also examine certain restricted notions of lowness for isometric isomorphism with respect to fixed computable presentations, and, in this vein, we obtain classifications of the degrees that are low for isometric isomorphism with respect to the standard copies of certain Lebesgue spaces.