Algorithmically Optimal Outer Measures

2020-06-15

arXiv.org

AI Summary

This work bridges the gap between algorithmic fractal dimension—rooted in computability theory—and classical local fractal dimensions from geometric measure theory. To achieve this, we introduce a Kolmogorov complexity-driven lower-semicomputable outer measure and rigorously establish, for the first time, the existence of a globally optimal outer measure. We further prove that any locally optimal outer measure induces a classical local dimension that coincides exactly with the algorithmic fractal dimension, thereby establishing their equivalence. This yields a rigorous unification of algorithmic and classical notions of fractal dimension. As a key contribution, we construct an explicit, convenient locally optimal outer measure κ, providing a novel measure-theoretic foundation for fractal geometry, effective randomness, and information theory. Moreover, our framework extends the applicability of the point-to-set principle to broader analytic and computational contexts.

Abstract

We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measure $oldsymbol{kappa}$ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

Problem

• Fractal Dimension
• Shape Boundary Measurement
• Optimal Algorithm

Innovation

• Fractal Dimensions
• Algorithmic Optimality
• Information Content Measurement