DeepAI AI Chat
Log In Sign Up

Algorithmically Optimal Outer Measures

06/15/2020
by   Jack H. Lutz, et al.
0

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 κ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

READ FULL TEXT

page 1

page 2

page 3

page 4

01/27/2021

Optimal Oracles for Point-to-Set Principles

The point-to-set principle <cit.> characterizes the Hausdorff dimension ...
07/28/2020

Algorithmic Fractal Dimensions in Geometric Measure Theory

The development of algorithmic fractal dimensions in this century has ha...
08/30/2021

An outer totalistic weakly universal cellular automaton in the dodecagrid with four states

In this paper, we prove that there is an outer totalistic weakly univers...
08/12/2019

Elements of asymptotic theory with outer probability measures

Outer measures can be used for statistical inference in place of probabi...
02/04/2021

Formalized Haar Measure

We describe the formalization of the existence and uniqueness of Haar me...
02/08/2022

Outer approximations of classical multi-network correlations

We propose a framework, named the postselected inflation framework, to o...
11/28/2017

TRPL+K: Thick-Restart Preconditioned Lanczos+K Method for Large Symmetric Eigenvalue Problems

The Lanczos method is one of the standard approaches for computing a few...