Algorithmically Optimal Outer Measures
Jack H. Lutz, Neil Lutz
TL;DR
The paper formalizes global and local optimality notions for lower semicomputable outer measures and proves the existence of globally optimal measures, along with a central theorem that every locally optimal outer measure $\mu$ on $\mathbb{R}^n$ has local fractal dimensions matching the algorithmic dimensions: $\mathrm{dim_{loc}} μ(x)=\dim(x)$ and $\mathrm{Dim_{loc}} μ(x)=\mathrm{Dim}(x)$ for all $x$. It centers the κ measure, defined by $\kappa(E)=2^{-\mathop{K}(E)}$, as a bridge between algorithmic and classical dimensions, and shows κ is locally optimal and not globally optimal, while constructing globally optimal examples (notably via a Levin-style lifting and a universal $\theta$). The work then derives point-to-set principles showing how relativized algorithmic dimensions characterize Hausdorff and packing dimensions through families of measures like $\kappa^A$, linking local coverings to global fractal structure. Together, these results provide a unified framework connecting algorithmic information densities with classical geometric measures in Euclidean spaces.
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 $\boldsymbolκ$ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.