A point to set principle for finite-state dimension

Elvira Mayordomo

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arXiv · 2208.00157

A point to set principle for finite-state dimension

Elvira Mayordomo

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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.

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Elvira Mayordomo. 2022-07-30. A point to set principle for finite-state dimension. https://arxiv.org/abs/2208.00157

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