A point to set principle for finite-state dimension
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916606867668992 |
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| author | Mayordomo, Elvira |
| author_facet | Mayordomo, Elvira |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_00157 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A point to set principle for finite-state dimension Mayordomo, Elvira Computational Complexity F.1.3; F.1.1 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. |
| title | A point to set principle for finite-state dimension |
| topic | Computational Complexity F.1.3; F.1.1 |
| url | https://arxiv.org/abs/2208.00157 |