Non-splittings of speedable sets

TL;DR

This paper constructs a speedable recursively enumerable set that cannot be split into two speedable sets, answering a question about the structural properties of speedable sets in computability theory.

Contribution

It provides the first example of a speedable set that defies splitting into speedable parts, advancing understanding of the internal structure of speedable sets.

Findings

Constructed a non-splittable speedable set

Solved a question posed by Bäuerle and Remmel

Demonstrated limitations of splitting speedable sets

Abstract

An r.e. set $A$ is speedable if for every recursive function, there exists a program enumerating membership in $A$ faster, by the desired recursive factor, on infinitely many integers. We construct a speedable set that cannot be split into speedable sets. This solves a question of B\"{a}uerle and Remmel.