The Point-to-Set Principle and the Dimensions of Hamel Bases

2021/09/22 by Jack H. Lutz, Lutz, Jack H., Renrui Qi +3 · 3 citations
Computer Science · Mathematics · #03D62 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #FOS: Mathematics #Logic (math.LO) #Logic in Computer Science (cs.LO) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2109.10981

openalex publication_date 2021/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that every real number in [0,1] is the Hausdorff dimension of a Hamel basis of the vector space of reals over the field of rationals. The logic of our proof is of particular interest. The statement of our theorem is classical; it does not involve the theory of computing. However, our proof makes essential use of algorithmic fractal dimension--a computability-theoretic construct--and the point-to-set principle of J. Lutz and N. Lutz (2018).

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