Does every $\alpha\in[0,1]$ arise as the Hausdorff dimension of a subring or subfield of $\mathbb{R}$? (Erdős #1154)
Statement
For each real $\alpha\in[0,1]$, does there exist a subring, or a subfield, of $\mathbb{R}$ whose Hausdorff dimension equals $\alpha$? Here a subring is a subset of $\mathbb{R}$ closed under addition, subtraction and multiplication (a subfield is additionally closed under division by nonzero elements), and its Hausdorff dimension is the usual metric dimension $\dim_H$. The question asks whether the full interval $[0,1]$ of dimensions is realised by such algebraic substructures of the real line.
Acceptance. FULLY RESOLVES: a complete proof (fully written; machine-checkable preferred where feasible) that in ZFC, for every $\alpha\in[0,1]$, there is a subring (resp. subfield) of $\mathbb{R}$ of Hausdorff dimension exactly $\alpha$ — with explicit control of $\dim_H$ — OR a proof that this statement is independent of ZFC (e.g. a model of ZFC in which some $\alpha\in[0,1]$ is not realised, complementing Mauldin's CH construction), OR a ZFC proof that some specific $\alpha$ cannot be realised. ADVANCES (each fully proved): realise the full interval for subrings under a hypothesis weaker than CH, or remove CH from Mauldin's subfield result for an explicit subinterval of $\alpha$; strengthen the Edgar–Miller/Falconer descriptive-set-theoretic obstructions (e.g. rule out further definability classes for intermediate-dimensional subrings); or construct, in ZFC, a subring/subfield of some intermediate dimension $\alpha\in(0,1)$. Deliver the construction with its dimension certificate, the independence proof with both models, or the impossibility/obstruction argument.
Background
Asked by Erdős [Er79h, p.119] and recorded as problem 2.48 in [Va99]; listed on erdosproblems.com/1154 (fetched 2026-07-21) with certificate class 'not disprovable' — open in general, but true in some models of set theory. The frontier is rich. Erdős and Volkmann [ErVo66] proved that for every $\alpha\in[0,1]$ there is an additive subgroup of $\mathbb{R}$ of Hausdorff dimension $\alpha$, leaving the harder ring/field case open. Strong regularity obstructions are known: Falconer [Fa84] showed a subring of Hausdorff dimension $\alpha\in(1/2,1)$ cannot be Borel or Suslin; Edgar and Miller [EdMi01] proved any real-closed analytic subfield of $\mathbb{R}$ has dimension $0$ or $1$, and [EdMi03] proved any Borel or analytic subring has dimension $0$ or is all of $\mathbb{R}$ — so any exotic intermediate-dimensional subring/subfield must be non-measurable (a 'wild' object built with choice). On the other hand Mauldin [Ma16b] proved that, assuming the continuum hypothesis, subfields of $\mathbb{R}$ of every Hausdorff dimension $\alpha\in[0,1]$ do exist, which is exactly why the status is 'not disprovable': the construction succeeds under CH, and no ZFC construction or refutation is known. No Erdős prize is attached. The attacker's tool is geometric measure theory plus set theory: transfinite/CH constructions of non-measurable fields with prescribed dimension, potential-theoretic and Fourier-dimension estimates to control $\dim_H$ of the generated ring, and forcing/absoluteness arguments to decide whether ZFC alone suffices.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1154 (T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.