We prove that the Continuum Hypothesis implies 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).

中文翻译：

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点对集原理、连续统假设和 Hamel 基的维数

我们证明连续统假设意味着 (0,1] 中的每个实数都是有理数域上实数向量空间的 Hamel 基的 Hausdorff 维数。我们证明的逻辑特别有趣。我们的定理是经典的；它不涉及计算理论。然而，我们的证明充分利用了算法分形维数——一种可计算性理论结构——以及 J. Lutz 和 N. Lutz 的点对集原则(2018)。