Edgar 2003 borel subrings reals

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theorem_1: Edgar and Miller's main theorem: a subring of the reals that is a Borel set either has Cartesian-Hausdorff dimension zero, so that it and all its finite Cartesian powers have Hausdorff dimension zero, or is the whole real line; the paper's remarks extend it to analytic sets.

theorem_2: The complex analog of Edgar and Miller's main theorem: a subring of the complex numbers that is a Borel set has Cartesian-Hausdorff dimension zero or equals the reals or the complex numbers.

theorem_3: The p-adic analog of Edgar and Miller's main theorem: a subring of the p-adic numbers that is a Borel set has Cartesian-Hausdorff dimension zero or equals the p-adic numbers or the p-adic integers; the proof is sketched.

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G. A. Edgar and Chris Miller, Borel subrings of the reals, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1121--1129; DOI 10.1090/S0002-9939-02-06653-4 (volume, issue and DOI from the Crossref record; the problem page gives "(2003), 1121-1129").

The copy read for this card is the authors' TeX preprint (dvipdfm, nine pages numbered 1--9), dated November 9, 2001 and marked "To appear in Proc. Amer. Math. Soc."; its text layer is clean. The journal version was not compared; page numbers and labels below are the preprint's. Provenance: downloaded in September 2026; the download URL was not recorded; 172,980 bytes. It is the authors' nine-page preprint, not the journal edition, and prints no copyright or license line. The AMS copyright policy (https://www.ams.org/publications/authors/ctp, read 2026-10-07) lets the authors reproduce a draft or accepted manuscript under CC BY for educational and scientific purposes, provided no commercial use is made and no derivative works are included, but it licenses no particular copy, and this file's download location was not recorded; the term is unstated.

Read status: claims checked. Theorem 1 with Lemmas 1.1--1.4, the Remarks extending it to analytic sets, Theorems 2 and 3 with the statements of Lemmas 2.1--2.4 and 3.1--3.4, and the Remarks of pp. 7--8 were read clause by clause on the page images (pp. 1--8); the proofs were not checked.

Contents

$\dim$ is Hausdorff dimension. By the Dimension Inequality $\dim(A\times B)\ge\dim A+\dim B$ for Borel sets, $(1/n)\dim(A^n)$ converges; the limit is the Cartesian--Hausdorff (CH) dimension of $A$, and it is $0$ iff $\dim(A^n)=0$ for every $n$, which forces $\dim A=0$ (p. 1).

• Theorem 1 (p. 1; lemmas and proof pp. 2--3): a subring of $\mathbb R$ that is a Borel set is either all of $\mathbb R$ or of CH dimension zero. Lemma 1.1 (p. 2): a Borel $A\subset\mathbb R^k$ with $\dim A>1$ has an image of positive Lebesgue measure under almost all linear functionals (the Projection Theorem). Lemma 1.2 (p. 2): a Borel additive subgroup $E$ of nonzero CH dimension has $\varphi(E^k)=\mathbb R$ for some $k$ and linear functional $\varphi$ (Steinhaus's theorem). Lemma 1.3 (pp. 2--3): for a subring $E$, such $k$ and $\varphi$ can be chosen with $\varphi$ injective on $E^k$ (take $k$ minimal). Lemma 1.4 (p. 3): if a linear functional maps $E^k$ bijectively onto $\mathbb R$ for a Borel additive subgroup $E$, then $k=1$ and $E=\mathbb R$ (a Borel measurable additive map $\mathbb R\to\mathbb R$ is linear).
• Remarks (pp. 3--4): Theorem 1 holds for analytic sets as well, since the Dimension Inequality, the Projection Theorem and Borel isomorphism extend to them; hence the ring $\mathbb Z[X]$ generated by an analytic set $X\subseteq\mathbb R$ with $\dim X^k>0$ for some $k$ is all of $\mathbb R$.
• Historical remarks (p. 2): Volkmann 1960 on subfields; Erdős and Volkmann 1966, for every $s\in[0,1]$ a Borel additive subgroup of dimension $s$ (card); Davies (unpublished, under the Continuum Hypothesis), subrings of every dimension $s\in[0,1]$, not Borel and not analytic; Falconer 1984, a Borel or analytic subring has $\dim E\le1/2$ or $\dim E=1$; the authors' result on real closed subfields (Edgar and Miller 2001).
• Divisible groups (p. 4): the divisible hull $\bigcup_n(1/n!)G$ of a Borel additive subgroup $G$ is a Borel $\mathbb Q$-vector space of the same dimension, so the Erdős--Volkmann groups give divisible Borel subgroups of every dimension in $[0,1]$.
• Theorem 2 (section 2, p. 4; lemmas pp. 4--7): a Borel subring $E\subset\mathbb C$ has zero CH dimension or $E=\mathbb R$ or $E=\mathbb C$. Theorem 3 (section 3, p. 7; proof sketched pp. 7--8): a Borel subring $E\subset\mathbb Q_p$ has zero CH dimension or $E=\mathbb Q_p$ or $E=\mathbb Z_p$.

Compiled scope

The statements above were checked on the page images and the proofs of Theorems 1 and 2 were read for structure; Theorem 3's proof is only sketched in the paper. No proof was checked in detail, and nothing here is independently reviewed. Result pages: Theorem 1, Theorem 2, Theorem 3.

Bears on. #1154, which asks for a ring or field in $\mathbb R$ of each Hausdorff dimension $\alpha\in[0,1]$: Theorem 1 and its analytic extension show that no Borel or analytic subring of $\mathbb R$, and so no such subfield, has Hausdorff dimension strictly between $0$ and $1$, so any example for $0<\alpha<1$ must lie outside those classes; the subrings of Davies's unpublished construction under the Continuum Hypothesis, which the paper reports (p. 2), are not Borel and not analytic. The theorem decides no $\alpha$ in $(0,1)$.

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