A point to set principle for finite-state dimension
Elvira Mayordomo
TL;DR
The paper develops a Euclidean-space characterization of base-$b$ finite-state dimension via the base-$b$ information content at precision and introduces a robust relativization through separator enumerators (SE). It proves a point-to-set principle, $\text{dim}_{\text{H}}(A)=\min_f\text{dim}_{\text{FS}}^f(A)$, linking Hausdorff dimension to relativized finite-state dimension and enabling geometric results to be obtained through finite-state methods. It further shows that $b$-normality corresponds to $\text{dim}_{\text{FS}}^b(x)=1$, tying normality to a finite-state informational notion in $[0,1)$. The work paves the way for extending fractal-dimensional results to relativized finite-state settings and raises open questions on the equidistribution properties of $f$-normality.
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.