Benign approximations and non-speedability

TL;DR

This paper explores the relationships between different classes of left-computable numbers, introducing new examples and counterexamples to clarify their properties and answering an open question about randomness and speedability.

Contribution

It constructs specific examples of left-computable numbers to demonstrate the incomparability of regainingly approximable and nearly computable classes, and resolves an open question on the link between randomness and speedability.

Findings

A non-computable number that is both regainingly approximable and nearly computable.

A left-computable number that is nearly computable but not regainingly approximable.

Speedable and regainingly approximable numbers are equivalent within nearly computable numbers.

Abstract

A left-computable number $x$ is called regainingly approximable if there is a computable increasing sequence $(x_n)_n$ of rational numbers converging to $x$ such that $x - x_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$; and it is called nearly computable if there is such an $(x_n)_n$ such that for every computable increasing function $s \colon \mathbb{N} \to \mathbb{N}$ the sequence ${(x_{s(n+1)} - x_{s(n)})_n}$ converges computably to 0. In this article we study the relationship between both concepts by constructing on the one hand a non-computable number that is both regainingly approximable and nearly computable, and on the other hand a left-computable number that is nearly computable but not regainingly approximable; it then easily follows that the two notions are incomparable with non-trivial intersection. With this relationship clarified, we then hold the keys to answering an open question of Merkle and Titov: they studied speedable numbers, that is, left-computable numbers whose approximations can be sped up in a certain sense, and asked whether, among the left-computable numbers, being Martin-L\"of random is equivalent to being non-speedable. As we show that the concepts of speedable and regainingly approximable numbers are equivalent within the nearly computable numbers, our second construction provides a negative answer.