Randomness versus superspeedability

TL;DR

This paper introduces superspeedable numbers, a subclass of speedable numbers with uniformly accelerable approximations, and explores their relationship with algorithmic randomness hierarchies.

Contribution

It defines superspeedable numbers, separates them from speedable numbers, and connects this hierarchy with the theory of algorithmic randomness.

Findings

Superspeedable numbers allow uniform acceleration of approximations.

A strict hierarchy of speedable and superspeedable numbers is established.

The hierarchy is related to existing notions of algorithmic randomness.

Abstract

Speedable numbers are real numbers which are algorithmically approximable from below and whose approximations can be accelerated nonuniformly. We begin this article by answering a question of Barmpalias by separating a strict subclass that we will refer to as superspeedable from the speedable numbers; for elements of this subclass, acceleration is possible uniformly and to an even higher degree. This new type of benign left-approximations of numbers then integrates itself into a hierarchy of other such notions studied in a growing body of recent work. We add a new perspective to this study by juxtaposing this hierachy with the well-studied hierachy of algorithmic randomness notions.