Regainingly approximable numbers and sets

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 $K(\alpha \upharpoonright n)>n$ 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.