The Point-to-Set Principle and the dimensions of Hamel bases

Article type: Research Article

Authors: Lutz, Jack H.^{a; *; ***} | Qi, Renrui^b | Yu, Liang^{c; **}

Affiliations: [a] Department of Computer Science, Iowa State University, Ames, IA 50011, USA | [b] School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand | [c] Department of Mathematics, Nanjing University, Jiangsu Province 210093, P.R. of China

Correspondence: [***] Corresponding author. [email protected].

Note: [*] Research supported in part by National Science Foundation grants 1545028 and 1900716.

Note: [**] Research supported in part by the National Science Foundation of China, No. 12025103.

Abstract: We prove that, for every 0⩽s⩽1, 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).

Keywords: Algorithmic fractal dimension, Hamel bases, Hausdorff dimension, Point-to-Set Principle

DOI: 10.3233/COM-210383

Journal: Computability, vol. 13, no. 2, pp. 105-112, 2024