On a conjecture of Dobrinen and Simpson concerning almost everywhere domination

TL;DR

This paper investigates the properties of almost everywhere domination in recursion theory, revealing its implications for randomness levels and establishing connections to notions like genericity.

Contribution

It provides a recursion-theoretic analysis of a.e. domination, linking it to randomness and showing that a.e. dominating sets have high randomness complexity.

Findings

Every a.e. dominating set makes 1-Z-randoms also 2-random.

Every a.e. dominating set satisfies 0' ≤ LR Z, indicating high randomness complexity.

The paper establishes new connections between a.e. domination, randomness, and genericity.

Abstract

The notions of almost everywhere (a.e.) domination and its uniform version were introduced and studied in reverse mathematics. This paper studies these notions from a recursion-theoretic point of view and explore their connections to notions such as randomness and genericity. It is shown that if $Z$ is a.e. dominating then each $1$-$Z$-random is $2$-random. In other words, $0'\leq_{\rm LR} Z$ for every a.e. dominating $Z$, where ${\rm LR}$ denotes low-for-random reducibility. Other results and corollaries are also given.