Problem 1154

Does there exist, for every $\alpha \in [0,1]$ , a ring or field in $\mathbb{R}$ with Hausdorff dimension $\alpha$ ?

Analysis

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem NOT DISPROVABLE, crediting the label to Mauldin's theorem that, assuming the continuum hypothesis, every $\alpha \in [0,1]$ is the Hausdorff dimension of a subfield of $\mathbb{R}$ , so that ZFC cannot refute the existence asked about unless ZFC is inconsistent. The claim page Mauldin's subfields of every dimension under CH records that consistency result as accepted. It settles one side only: whether ZFC alone proves the existence is not settled. This page departs from the site's label because one side alone leaves the question open; a matching result that ZFC does not prove the existence would settle the problem as independent.

EdMi01: Edgar, G. A. and Miller, Chris, Hausdorff dimension, analytic sets and transcendence. Real Anal. Exchange (2001/02), 335-339.
EdMi03: Edgar, G. A. and Miller, Chris, Borel subrings of the reals. Proc. Amer. Math. Soc. (2003), 1121-1129.
Bo03: Bourgain, J., On the Erdős-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal. 13 (2003), no. 2, 334-365. DOI.
ErVo66: Erdős, Paul and Volkmann, Bodo, Additive Gruppen mit vorgegebener Hausdorffscher Dimension. J. Reine Angew. Math. (1966), 203-208.
Fa84: Falconer, K. J., Rings of fractional dimension. Mathematika (1984), 25-27.
Ma16b: Mauldin, R. Daniel, Subfields of $\mathbf{R}$ with arbitrary Hausdorff dimension. Math. Proc. Cambridge Philos. Soc. 161 (2016), no. 1, 157-165.

Problem 1154

Status

Proof claims

Discussion