Effective bi-immunity and randomness

TL;DR

This paper explores the connection between randomness and effective bi-immunity, demonstrating that certain random sequences do not compute effectively bi-immune sets and highlighting key differences between these properties.

Contribution

It extends previous results by showing the same properties hold for effective bi-immunity and identifies the class of oracles where ML randomness implies effective bi-immunity.

Findings

Existence of sequences with effective Hausdorff dimension 1 that do not compute effectively bi-immune sets.

The class Low(MLR, EBI) includes jump-traceable sets and has continuum cardinality.

Abstract

We study the relationship between randomness and effective bi-immunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing DNR functions relative to X that compute no ML random set. We show that the same holds when ML randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set. We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every ML random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.