Benign approximations and non-speedability

A left-computable number \(x\) is called regainingly approximable if there is a computable increasing sequence \((x_n)_n\) of rational numbers converging to \(x\) such that \(x - x_n < 2^{-n}\) for infinitely many \(n \in \mathbb{N}\); and it is called nearly computable if there is such an \((x_n)_n\) such that for every computable increasing function \(s \colon \mathbb{N} \to \mathbb{N}\) the sequence \({(x_{s(n+1)} - x_{s(n)})_n}\) converges computably to 0. In this article 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 non-trivial 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-L\"of 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.