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The Point-to-Set Principle and the Dimensions of Hamel Bases (2109.10981v4)
Published 22 Sep 2021 in cs.LO and math.LO
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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