Algorithmically Optimal Outer Measures

TL;DR

This paper explores the connection between algorithmic and classical fractal dimensions, proving the existence of globally optimal outer measures and their equivalence in local fractal dimensions.

Contribution

It introduces global and local optimality conditions for outer measures and proves the existence of globally optimal measures with dimensions matching algorithmic fractal dimensions.

Findings

Globally optimal outer measures exist.

Classical local fractal dimensions of locally optimal measures match algorithmic dimensions.

Uses Kolmogorov complexity to define a convenient optimal outer measure.

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 $\boldsymbol{\kappa}$ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.