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.

Abstract A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number x is reordered computable if there exist a computable function $$f :\mathbb {N} \rightarrow \mathbb {N}$$ f : N → N with $$\sum _{k=0}^{\infty } 2^{-f(k)} = x$$ ∑ k = 0 ∞ 2 - f ( k ) = x and a bijective function $$\sigma :\mathbb {N} \rightarrow \mathbb {N}$$ σ : N → N such that the rearranged series $$\sum _{k=0}^{\infty } 2^{-f(\sigma (k))}$$ ∑ k = 0 ∞ 2 - f ( σ ( k ) ) converges computably. In this article we will give some examples and counterexamples for reordered computable numbers and we will show that these numbers are closed under addition, multiplication and the Solovay reduction. Finally, we will also present a density theorem for reordered computable numbers.