Hausdorff dimension of the class of minimal Turing degrees (Min)

Determine the classical Hausdorff dimension of the class Min = {X ∈ 2^ω : X has minimal Turing degree}; moreover, ascertain whether the effective Hausdorff dimension of Min equals 1.

Background

The paper connects the Hausdorff and packing dimensions of maximal antichains in the Turing degrees to properties of minimal degrees. Using Downey and Greenberg’s theorem on packing dimension, the authors show that every maximal antichain has packing dimension 1 and relate potential Hausdorff dimension bounds for maximal antichains to the Hausdorff dimension of the class of minimal degrees (Min).

Despite progress on dimensions and randomness within degree structures, the precise Hausdorff dimension of Min is unknown. This question is tied to ongoing research on diagonally non-recursive functions, forcing with bushy trees, and properties of minimal degrees, with partial results suggesting potential high complexity but no definitive determination.