Dimension of Generic Reals

Abstract

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

Figures (3)

• Figure :
• Figure :
• Figure :

Theorems & Definitions (43)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 2.1
• Definition 2.2
• Proposition 2.1
• Proposition 2.2
• Corollary 2.2.1
• Proposition 2.3
• Definition 3.1