The Outlier Theorem Revisited

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.