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arXiv · 2603.08952

Dimension of Generic Reals

Yiping Miao
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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.

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Yiping Miao. 2026-03-09. Dimension of Generic Reals. https://arxiv.org/abs/2603.08952

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