A point to set principle for finite-state dimension

2022-07-30

arXiv.org

AI Summary

This paper addresses the lack of an information-theoretic characterization and a point-to-set principle for finite-state dimension (FS dimension). It introduces a precise quantification of the information content of real numbers under finite precision, integrating effective dimension theory, finite-state automaton complexity, and relativized algorithmic information theory. The work establishes, for the first time, a point-to-set principle for FS dimension and thereby defines a robust notion of relative normality. It then rigorously proves the equivalence between relative normality and FS dimension. The main contributions are: (1) an information-theoretic, exact definition of FS dimension; (2) a necessary and sufficient characterization linking relative normality to FS dimension; and (3) a foundational framework enabling future investigation of its equidistribution properties.

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.

Problem

• Characterize finite-state dimension via information content.
• Develop a robust concept of relativized normality.
• Prove a finite-state dimension point-to-set principle.

Innovation

• Finite-state dimension characterization
• Relativized normality robust concept
• Point-to-set principle application