The Outlier Theorem Revisited

TL;DR

This paper revisits the outlier theorem in algorithmic information theory, providing a simpler proof with improved bounds, extending it to dynamical systems, and exploring implications for computability and randomness.

Contribution

It offers a simplified proof of the outlier theorem, extends it to ergodic systems, and discusses the computability and information-theoretic properties of sets in the Cantor space.

Findings

Outliers must appear in computable sampling methods.

Extended the outlier theorem to ergodic dynamical systems.

Large measure open sets in Cantor space have either simple members or high mutual information.

Abstract

An outlier is a datapoint that is set apart from a sample population. The outlier theorem in algorithmic information theory states that given a computable sampling method, outliers must appear. We present a simple proof to the outlier theorem, with exponentially improved bounds. We extend the outlier theorem to ergodic dynamical systems which are guaranteed to hit ever larger outlier states with diminishing measures. We show how to construct deterministic functions from random ones, i.e. function derandomization. We also prove that all open sets of the Cantor space with large uniform measure will either have a simple computable member or high mutual information with the halting sequence.