A note on the differences of computably enumerable reals

TL;DR

The paper investigates properties of non-computable left-c.e. reals, showing that for any such real, there exists another with a specific non-decomposability property, highlighting differences in halting probabilities of universal machines.

Contribution

It demonstrates the existence of left-c.e. reals that cannot be expressed as sums involving other left- or right-c.e. reals, revealing new structural distinctions among computably enumerable reals.

Findings

Existence of left-c.e. reals not decomposable as sums with other c.e. reals.

Existence of universal machines with halting probabilities not related by left-c.e. translations.

The proof depends on whether the real is Martin-Löf random, indicating a dichotomy.

Abstract

We show that given any non-computable left-c.e. real $\alpha$ there exists a left-c.e. real $\beta$ such that $\alpha\neq \beta+\gamma$ for all left-c.e. reals and all right-c.e. reals $\gamma$. The proof is non-uniform, the dichotomy being whether the given real $\alpha$ is Martin-Loef random or not. It follows that given any universal machine $U$, there is another universal machine $V$ such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact.