Kolmogorov Complexity of Attractive Degrees

TL;DR

The paper studies attractive versus dispersive Turing degrees in the $H$-Hausdorff framework for coarse computation and introduces a Kolmogorov-complexity flavored sufficient condition for attractivity. It proves that if for some constant $C$ we have $K(A \upharpoonright 2^n) > 2K(n) - C$ for all large $n$, then $A$ is attractive, and generalizes with a bound $K(A \upharpoonright 2^n) \ge K(n) + K^X(n) - C$ using an oracle $X$. The results illuminate the relationship between randomness and genericity by showing $1$-random sets are attractive and by deriving consequences for $1$-generic sets, and they establish via the point-to-set principle that the dispersive class has classical Hausdorff dimension $0$. They also prove the existence of minimal dispersive degrees and discuss why the approach does not automatically yield minimal attractive degrees, posing open questions about minimal degrees with $K(A \upharpoonright n) > 2K(n)$ for all $n$.

Abstract

This paper proves a Kolmogorov-complexity-flavored sufficient condition for a set to be attractive and discusses some consequences of this condition.

Theorems & Definitions (14)

• Definition 1.1
• Theorem 1.2: Monin2018metricpdf
• Theorem 1.3: Monin2018metricpdf
• Definition 1.4: metricpdf
• Proposition 1.5: metricpdf
• proof
• Theorem 2.1
• proof
• Corollary 2.2: of the proof
• Proposition 2.3