Dimension of Generic Reals

TL;DR

This paper explores the measure-theoretic properties of various classes of generic reals in computability theory, revealing conditions under which these sets have positive measure based on gauge functions and Turing ideals.

Contribution

It establishes precise measure conditions for different types of generic reals, connecting gauge functions with the structure of Turing ideals in computability theory.

Findings

$ ext{Gamma}$-Cohen generics have positive measure iff gauge function is not dominated by all in $ ext{Gamma}$

$ ext{Gamma}$-Mathias and $ ext{Gamma}$-Sacks generics have positive measure iff gauge function eventually dominates all in $ ext{Gamma}$

Provides comparison between the measure of generic sets and their computational properties

Abstract

This paper investigates the Hausdorff measure of certain sets of generics in computability theory. Let $\Gamma$ be the Turing ideal in which we take the dense open sets. The set of $\Gamma$-Cohen generics has measure positive if and only if the gauge function is not dominated by every element in $\Gamma$, under some mild restrictions on the gauge function. The set of $\Gamma$-Mathias generics and the set of $\Gamma$-Sacks generics have measure positive if and only if the gauge function eventually dominates every element in $\Gamma$. This gives some comparison between the behavior of reals in the set and the measure of the set.