Oracle-relativized effective dimension of maximal antichains of Turing degrees

Determine whether, for every oracle Z ∈ 2^ω, every maximal antichain of Turing degrees has effective-in-Z Hausdorff dimension 1; if not, identify the minimal complexity requirement on Z such that there exists a maximal antichain of Turing degrees whose effective-in-Z Hausdorff dimension is strictly less than 1, and decide whether Z = ∅′ suffices.

Background

The authors prove that any maximal antichain of Turing degrees has effective Hausdorff dimension 1, but this result does not relativize in general; it holds relative to K-trivial oracles due to low-for-randomness properties. They discuss obstacles to broader relativization, including the failure of partial relativization of certain randomness-based arguments and the low effective dimension of hyperimmune-free reals.

This motivates an inquiry into whether stronger oracles can reduce the effective dimension of maximal antichains below 1. The problem seeks a complete characterization of the oracle complexity necessary for such reductions and specifically asks whether the halting problem (∅′) is sufficient.

References

This tension compels us to formalize the following open problem: Is it true that given any oracle $Z$, any maximal antichain of Turing degrees has effective-in-$Z$ Hausdorff dimension 1? If not, in what complexity does the oracle have to be so that there is some maximal antichain of Turing degrees that has effective-in-$Z$ Hausdorff dimension $<1$? Can $\emptyset'$ do the job?

— On the Hausdorff dimension of maximal chains and antichains of Turing and Hyperarithmetic degrees (2504.04957 - Song et al., 7 Apr 2025) in Section 5 (Antichains in Turing degrees), end; Question environment