A point to set principle for finite-state dimension

07/30/2022

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by   Elvira Mayordomo, et al.

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Effective dimension has proven very useful in geometric measure theory through the point-to-set principle <cit.> that characterizes Hausdorff dimension by relativized effective dimension. Finite-state dimension is the least demanding effectivization in this context <cit.> that among other results can be used to characterize Borel normality <cit.>. 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 prove a finite-state dimension point-to-set principle.