Speedable Left-c.e. Numbers

Abstract

A left-c.e. real number $\alpha$ is $\rho$-speedable if there is a computable left approximation $a_0, a_1, \ldots$ of $\alpha$ and a nondecreasing computable function f such that we have $f(n) \ge n$ and

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\frac{\alpha -a_{f(n)}}{\alpha -a_n}\le \rho , \end{aligned}$$

and $\alpha$ is speedable if it is $\rho$-speedable for some $\rho <1$. Barmpalias and Lewis-Pye [JCSS 89:349–360, 2016] have implicitly shown that Martin-Löf random left-c.e. real numbers are never speedable. We give a straightforward direct proof of this fact and state as open problem whether this implication can be reversed, i.e., whether all nonspeedable left c.e. real numbers are Martin-Löf random. In direction of solving the latter problem, we demonstrate that speedability is a degree property for Solovay degrees in the sense that either all or no real numbers in such a degree are speedable, and that left-c.e. real numbers of nonhigh Turing degree are always speedable. In this connection, we observe that every c.e. Turing degree contains a speedable left-c.e. real number. Furthermore, we obtain a dichotomy result: by definition, left-approximations of nonspeedable real numbers are never speedable, while for any speedable real number all of its left approximations are $\rho$-speedable for all $\rho >0$.

The second author was supported by Landesgraduiertenförderung Baden-Württemberg.