On the Hausdorff dimension of maximal chains and antichains of Turing and Hyperarithmetic degrees
Sirun Song, Liang Yu
TL;DR
This work analyzes fractal dimensions of degree structures in computability theory, focusing on maximal chains and maximal antichains in the Turing and hyperarithmetic lattices. It develops a toolkit including $\Pi^1_1$ randomness, $\Delta^1_1$-domination, recursive traceability, and Sacks forcing to quantify Hausdorff and packing dimensions of these degree configurations. The authors prove that all maximal antichains in hyperdegrees have Hausdorff dimension $1$, construct a maximal $\omega_1$-chain in the Turing degrees with effective Hausdorff dimension $0$ (and a $\Pi^1_1$-definable version under $\omega_1=(\omega_1)^L$), and analyze maximal antichains in Turing degrees—showing effective Hausdorff dimension $1$ (relativized to K-trivial oracles) and packing dimension $1$ as well. They also discuss relativization limits and pose open questions regarding the minimal possible dimension under various oracles.
Abstract
This paper investigates the Hausdorff dimension properties of chains and antichains in Turing degrees and hyperarithmetic degrees. Our main contributions are threefold: First, for antichains in hyperarithmetic degrees, we prove that every maximal antichain necessarily attains Hausdorff dimension 1. Second, regarding chains in Turing degrees, we establish the existence of a maximal chain with Hausdorff dimension 0. Furthermore, under the assumption that $ω_1=(ω_1)^L$, we demonstrate the existence of such maximal chains with $Π^1_1$ complexity. Third, we extend our investigation to maximal antichains of Turing degrees by analyzing both the packing dimension and effective Hausdorff dimension.