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.
Hölzl et al. (Tue,) studied this question.