A point to set principle for finite-state dimension

TL;DR

This paper introduces a new characterization of finite-state dimension based on information content, establishes a point-to-set principle for it, and explores implications for relativized normality and equidistribution properties.

Contribution

It provides a novel point-to-set principle for finite-state dimension and a robust framework for relativized normality based on information content.

Findings

Characterization of finite-state dimension via information content

Finite-state dimension point-to-set principle established

Open question on relativized normality and equidistribution

Abstract

Effective dimension has proven very useful in geometric measure theory through the point-to-set principle \cite{LuLu18}\ that characterizes Hausdorff dimension by relativized effective dimension. Finite-state dimension is the least demanding effectivization in this context \cite{FSD}\ that among other results can be used to characterize Borel normality \cite{BoHiVi05}. In this paper we prove a characterization of finite-state dimension in terms of information content of a real number at a certain precision. We then use this characterization to give a robust concept of relativized normality and prove a finite-state dimension point-to-set principle. We finish with an open question on the equidistribution properties of relativized normality.