A point to set principle for finite-state dimension
Computational Complexity
2025-02-12 v2
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.
Keywords
Cite
@article{arxiv.2208.00157,
title = {A point to set principle for finite-state dimension},
author = {Elvira Mayordomo},
journal= {arXiv preprint arXiv:2208.00157},
year = {2025}
}