The Point-to-Set Principle and the dimensions of Hamel bases
Jack H. Lutz, Renrui Qi, Renrui Qi, Liang Yu · Computability · 2024
We prove that, for every [Formula: see text], there is a Hamel basis of the vector space of reals over the field of rationals that has Hausdorff dimension s. 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).