Speedability of computably approximable reals and their approximations

2026-03-27

AI Summary

This study investigates the accelerability of computable approximations to real numbers, with a focus on difference-computably-enumerable (d.c.e.) reals. Employing tools from computable analysis, recursion theory, and algorithmic randomness—particularly constructions involving sequences of bounded variation and computable functions—the work extends the notion of accelerability equivalence beyond the classical left-c.e. setting to d.c.e. reals for the first time. It establishes that a d.c.e. real admits an accelerated computable approximation if and only if it is not Martin-Löf random. Furthermore, it proves that every computably approximable real possesses at least one approximating sequence that can be accelerated. These results forge a precise equivalence between accelerability and non-Martin-Löf randomness for d.c.e. reals, thereby significantly broadening the theoretical framework previously confined to left-c.e. cases.

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 $\{a_s\}_{s \in ω}$ is speedable if there exists a nondecreasing computable function $f$ such that the approximation $\{a_{f(s)}\}_{s \in ω}$ converges in a certain formal sense faster than $\{a_s\}_{s \in ω}$. 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-Löf random. Finally, we demonstrate that every computably approximable real has a computable approximation that is speedable.

Problem

• speedability
• computably approximable reals
• d.c.e. reals
• approximations
• Martin-Löf randomness

Innovation

• speedability
• computably approximable reals
• d.c.e. reals
• Martin-Löf randomness
• computable approximation