Regainingly approximable numbers and sets

TL;DR

This paper introduces the concept of regainingly approximable numbers and sets, exploring their properties, complexity, and relationships with computability and randomness, revealing their rich structure and limitations.

Contribution

It defines regainingly approximable numbers and sets, analyzes their properties, and demonstrates their distribution across Turing degrees and their complexity characteristics.

Findings

Regainingly approximable sets are not closed under union or intersection.

Every c.e. Turing degree contains a regainingly approximable set.

Regainingly approximable numbers are between computable and left-computable numbers, not closed under addition.

Abstract

We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We also call a set $A\subseteq\mathbb{N}$ regainingly approximable if it is c.e. and the strongly left-computable number $2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. $K$-trivial, we construct such an $\alpha$ such that ${K(\alpha \restriction n)>n}$ for infinitely many $n$. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.