We prove that, for every , 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).
T.M.Apostol, Calculus, Volume II: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability, John Wiley & Sons, 1969.
2.
A.Blass, Existence of bases implies the axiom of choice, Contemporary Mathematics31 (1984).
3.
C.Cabrelli, K.E.Hare and U.Molter, Sums of Cantor sets yielding an interval, Journal of the Australian Mathematical Society73 (2002), 405–418, https://api.semanticscholar.org/CorpusID:11396262. doi:10.1017/S1446788700009058.
4.
R.G.Downey and D.R.Hirschfeldt, Algorithmic Randomness and Complexity, Springer Science & Business Media, 2010.
G.Hamel, Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: , Mathematische Annalen60(3) (1905), 459–462. doi:10.1007/BF01457624.
8.
F.B.Jones, Measure and other properties of a Hamel basis, Bulletin of the American Mathematical Society48(6) (1942), 472–481. doi:10.1090/S0002-9904-1942-07710-X.
9.
M.Li and P.M.B.Vitányi, An Introduction to Kolmogorov Complexity and Its Applications, 4th edn, Springer Publishing Company, Incorporated, 2019. ISBN 3030112977.
10.
J.H.Lutz, The dimensions of individual strings and sequences, Information and Computation187(1) (2003), 49–79. doi:10.1016/S0890-5401(03)00187-1.
11.
J.H.Lutz and N.Lutz, Algorithmic information, plane Kakeya sets, and conditional dimension, ACM Transactions on Computation Theory (TOCT)10(2) (2018), 1–22. doi:10.1145/3201783.
12.
J.H.Lutz and N.Lutz, Who asked us? How the theory of computing answers questions about analysis, in: Complexity and Approximation, Springer, 2020, pp. 48–56. doi:10.1007/978-3-030-41672-0_4.
13.
J.H.Lutz and E.Mayordomo, Dimensions of points in self-similar fractals, SIAM Journal on Computing38(3) (2008), 1080–1112. doi:10.1137/070684689.
14.
J.H.Lutz and E.Mayordomo, Algorithmic fractal dimensions in geometric measure theory, in: Handbook of Computability and Complexity in Analysis, Springer, 2021, pp. 271–302.
15.
N.Lutz, Fractal intersections and products via algorithmic dimension, ACM Trans. Comput. Theory13(3) (2021), 14:1–14:15. doi:10.1145/3460948.
16.
N.Lutz and D.M.Stull, Projection theorems using effective dimension, in: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Schloss Dagstuhl – Leibniz-Zentrum fuer Informatik, 2018.
17.
N.Lutz and D.M.Stull, Bounding the dimension of points on a line, Information and Computation275 (2020), 104601. doi:10.1016/j.ic.2020.104601.
18.
E.Mayordomo, A Kolmogorov complexity characterization of constructive Hausdorff dimension, Information Processing Letters84(1) (2002), 1–3. doi:10.1016/S0020-0190(02)00343-5.
19.
Y.Moschovakis, Notes on Set Theory, Springer, 2005.
20.
T.Orponen, Combinatorial proofs of two theorems of Lutz and Stull, in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 2021, pp. 1–12.
21.
J.Roitman, Introduction to Modern Set Theory, John Wiley & Sons, 1990.
22.
A.Shen, V.A.Uspensky and N.Vereshchagin, Kolmogorov Complexity and Algorithmic Randomness, Vol. 220, American Mathematical Soc., 2017.
23.
W.Sierpiński, Sur la question de la mesurabilité de la base de M. Hamel, Fundamenta Mathematicae1(1) (1920), 105–111. doi:10.4064/fm-1-1-105-111.
24.
T.A.Slaman, On capacitability for co-analytic sets, New Zealand J. Math.52 (2021), 865–869. doi:10.1007/s13226-021-00101-z.
25.
D.M.Stull, Optimal oracles for Point-To-Set Principles, in: 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, Marseille, France, March 15–18, 2022, P.Berenbrink and B.Monmege, eds, LIPIcs, Vol. 219, Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2022, pp. 57:1–57:17. Virtual Conference. doi:10.4230/LIPIcs.STACS.2022.57.
