2022/07/30 by Elvira Mayordomo, Mayordomo, Elvira · 2 citations
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #F.1.1 #F.1.3 #FOS: Computer and information sciences #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2208.00157
openalex publication_date 2022/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Effective dimension has proven very useful in geometric measure theory through the point-to-set principle \citeLuLu18 that characterizes Hausdorff dimension by relativized effective dimension. Finite-state dimension is the least demanding effectivization in this context \citeFSD that among other results can be used to characterize Borel normality \citeBoHiVi05. 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.