Benign approximations and non-speedability

A left-computable number [Formula: see text] is called regainingly approximable if there is a computable increasing sequence [Formula: see text] of rational numbers converging to [Formula: see text] such that [Formula: see text] for infinitely many [Formula: see text]; and it is called nearly computable if there is such an [Formula: see text] such that for every computable increasing function [Formula: see text] the sequence [Formula: see text] converges computably to 0. In this paper, 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 nontrivial 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-Löf 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.