Edgar 2001 hausdorff dimension analytic sets transcendence

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theorem: An analytic real closed proper subfield of the reals has Hausdorff dimension zero; the proof shows that no analytic set of positive dimension lies in a proper real closed subfield.

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G. A. Edgar and Chris Miller, Hausdorff dimension, analytic sets and transcendence, Real Anal. Exchange 27 (2001/02), no. 1, 335--339. The problem page gives "(2001/02), 335-339"; the volume and issue are from the Crossref record read (UTC), which lists the article under the JSTOR DOI 10.2307/44154130.

The copy read for this card is the authors' TeX preprint (pdfTeX, four pages numbered 1--4), dated June 8, 2001 and marked "To appear in Real Analysis Exchange"; its text layer is clean. The journal version was not compared; page numbers below are the preprint's. Provenance: downloaded in September 2026; the download URL was not recorded; 141,894 bytes. It is the authors' four-page preprint, not the journal edition, and prints no copyright or license line; no publisher page applies to it and its download location was not recorded; the term is unstated.

Read status: claims checked. The Theorem and Lemmas 1--4 were read clause by clause in the text layer (pp. 2--3); the short proofs of Lemmas 1--3 and of the Theorem (pp. 2--4) were read but not checked, and Lemma 4 is stated without proof as a special case of a result of van den Dries.

Contents

"Dimension" means Hausdorff dimension $\dim_H$; analytic subsets of $\mathbb R^n$ are those obtained from Borel subsets of $\mathbb R$ by continuous maps; an ordered field is real closed when it contains square roots of its positive elements and a root of each odd-degree polynomial over it (p. 1).

• Theorem (p. 2; proof pp. 3--4): if $K\subsetneq\mathbb R$ is a real closed subfield and an analytic set, then $\dim_HK=0$. The proof gives more: an analytic $E\subset\mathbb R$ with $\dim_HE>0$ lies in no proper real closed subfield; equivalently, $E$ contains a transcendence base for $\mathbb R$ (abstract and p. 2). The converse fails: there are compact sets of dimension $0$ whose sum set has interior (p. 2).
• Lemma 1 (p. 2): for compact $E\subset\mathbb R$ with $\dim_HE>0$ there are $n$ and an $\mathbb R$-linear $T:\mathbb R^n\to\mathbb R$ such that $T(E^n)$ has interior (through $\dim_H(E^k)\ge k\dim_HE>1$, a projection with image of positive measure, and the difference-set theorem). Lemma 2 (p. 2): the same for analytic $E$. The Remark after Lemma 2 notes that for an analytic additive subgroup $E$ of positive dimension, $T(E^n)=\mathbb R$.
• Lemma 3 (p. 3): the smallest real closed subfield of $\mathbb R$ containing an analytic set is analytic (through countably many semialgebraic functions defined over $\mathbb Q$ and cell decomposition); the Remark says this fails with "Borel" in place of "analytic", though the real closure of a Borel subfield is Borel.
• Lemma 4 (p. 3, stated without proof as a special case of van den Dries, Fund. Math. 157 (1998), Lemma 4.1): if $K\subsetneq L$ are real closed subfields of $\mathbb R$ and $f:\mathbb R^n\to\mathbb R$ is semialgebraic and defined over $L$, then $f(K^n)$ has empty interior in $L$.
• Context (p. 2): every proper additive subgroup of $\mathbb R$ is cyclic or dense and co-dense; Erdős and Volkmann give, for each $d\in[0,1]$, a Borel subgroup of dimension $d$ (card); for Borel subrings the dimension was known to be $1$ or at most $1/2$, with no examples other than dimensions $0$ and $1$, and the subfield question was open; the note answers it for real closed subfields: a Borel or analytic one has dimension $0$ or $1$, and $1$ only for $\mathbb R$ itself. The authors' later paper (Edgar and Miller 2003) treats Borel subrings.

Compiled scope

The whole four-page note was read in the text layer; the statements were checked and the proofs were read but not checked. Nothing here is independently reviewed.

Bears on. #1154, which asks for a ring or field in $\mathbb R$ of each Hausdorff dimension $\alpha\in[0,1]$: the Theorem rules out real closed subfields of dimension strictly between $0$ and $1$ among analytic (in particular Borel) sets, while the problem does not restrict the ring or field to such sets.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.