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).

翻译：我们证明[0,1]中的每个实数都是有理数域上实数向量空间的某个哈梅尔基的豪斯多夫维数。我们证明的逻辑具有特殊意义：定理的表述是经典的，不涉及计算理论。然而，我们的证明本质性地运用了算法分形维数（一种可计算性理论构造）以及J. Lutz与N. Lutz（2018）提出的点集原理。