Speedability of computably approximable reals and their approximations

Abstract

An approximation of a real is a sequence of rational numbers that converges to the real. An approximation is left-c.e. if it is computable and nondecreasing and is d.c.e. if it is computable and has bounded variation. A real is computably approximable if it has some computable approximation, and left-c.e. and d.c.e. reals are defined accordingly. An approximation $\{a_s\}_{s \in \omega}$ is speedable if there exists a nondecreasing computable function $f$ such that the approximation $\{a_{f(s)}\}_{s \in \omega}$ converges in a certain formal sense faster than $\{a_s\}_{s \in \omega}$. This leads to various notions of speedability for reals, e.g., one may require for a computably approximable real that either all or some of its approximations of a specific type are speedable. Merkle and Titov established the equivalence of several speedability notions for left-c.e. reals that are defined in terms of left-c.e. approximations. We extend these results to d.c.e. reals and d.c.e. approximations, and we prove that in this setting, being speedable is equivalent to not being Martin-L\"{o}f random. Finally, we demonstrate that every computably approximable real has a computable approximation that is speedable.