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Statement

Theorem (p. 2). The paper states it as "Every analytic real closed proper subfield of $\mathbb R$ has dimension $0$", where "dimension" means Hausdorff dimension (p. 1).

Here a subset of $\mathbb R^n$ is analytic when it is the continuous image of a Borel subset of $\mathbb R$, and an ordered field is real closed when each of its positive elements has a square root in it and each odd-degree polynomial in one variable with coefficients in it has a root in it (p. 1). Every Borel subset of $\mathbb R^n$ is analytic (p. 1).

Stronger form proved (p. 2, proof pp. 3--4). If $E\subseteq\mathbb R$ is analytic and $\dim_HE>0$, then $E$ is contained in no proper real closed subfield of $\mathbb R$. The paper restates this as: $E$ contains a transcendence base for $\mathbb R$, a maximal algebraically independent subset of $\mathbb R$ (abstract and p. 2). The Theorem follows by taking $E=K$.

What it gives for subfields (p. 2). A real closed subfield of $\mathbb R$ that is a Borel set, or more generally an analytic set, has Hausdorff dimension $0$ or $1$, and dimension $1$ only when it is $\mathbb R$ itself.

The converse fails (p. 2). Dimension $0$ does not keep an analytic set inside a proper real closed subfield: there are compact $C\subseteq\mathbb R$ of dimension $0$ whose sum set $\{x+y:x,y\in C\}$ has interior, so $C$ lies in no proper additive subgroup of $\mathbb R$. The paper takes $C=E\cup F$ with $E,F$ from Falconer's Fractal geometry (1990), Example 7.8.

Source. G. A. Edgar and Chris Miller, Hausdorff dimension, analytic sets and transcendence, Real Anal. Exchange 27 (2001/02), no. 1, 335--339. Page numbers are those of the authors' four-page preprint identified on the source card; the journal edition was not compared.

Read depth. Claims checked: the statement, the stronger form, the four lemmas and the proof of the Theorem were read clause by clause. Lemma 4 is stated in the paper without proof, and the facts the lemmas cite from Mattila, Oxtoby, Edgar and real algebraic geometry were not re-derived. Nothing here is independently reviewed.

Proof pointer

Pages 2--4. The paper says the Theorem is immediate from four lemmas.

• Lemma 1 (p. 2): for compact $E\subseteq\mathbb R$ with $\dim_HE>0$ there are $n\in\mathbb N$ and an $\mathbb R$-linear $T:\mathbb R^n\to\mathbb R$ with $T(E^n)$ having interior in $\mathbb R$. One picks $k$ with $k\dim_HE>1$, so $\dim_H(E^k)>1$, takes an orthogonal projection $\pi:\mathbb R^k\to\mathbb R$ whose image of $E^k$ has positive Lebesgue measure, and uses that the difference set of a set of positive measure has interior; then $n=2k$.
• Lemma 2 (p. 2): the same conclusion for analytic $E\subseteq\mathbb R$ with $\dim_HE>0$, since such $E$ contains a compact set of positive dimension. The Remark after it notes that if $E$ is also an additive subgroup then $T(E^n)=\mathbb R$.
• Lemma 3 (p. 3): if $E\subseteq\mathbb R$ is analytic, the smallest real closed subfield of $\mathbb R$ containing $E$ is analytic. It is the union of the images $f(E^n)$ over the countably many semialgebraic $f:\mathbb R^n\to\mathbb R$ defined over $\mathbb Q$, and cell decomposition makes each image a finite union of continuous images of analytic sets. The Remark after it, credited to R. Dougherty, says the lemma fails with "Borel" in place of "analytic", even when $E$ is a subring, while the real closure of a Borel subfield is Borel.
• Lemma 4 (p. 3), a special case of van den Dries, Dense pairs of o-minimal structures, Fund. Math. 157 (1998), Lemma 4.1, stated without proof: if $K\subsetneq L$ are real closed subfields of $\mathbb R$, $n\in\mathbb N$, and $f:\mathbb R^n\to\mathbb R$ is semialgebraic and defined over $L$, then $f(K^n)$ has empty interior in $L$.
• Proof of the Theorem (pp. 3--4): for analytic $E$ with $\dim_HE>0$, the smallest real closed field $K\supseteq E$ is analytic by Lemma 3 and has positive dimension; Lemma 2 gives a linear, hence semialgebraic, $T$ with $T(K^n)$ having interior; Lemma 4 with $L=\mathbb R$ forces $K=\mathbb R$.

Dependencies

Lemmas 1--4 of the same paper. Through them: Mattila's Geometry of sets and measures in Euclidean spaces (1995) for the product-dimension bound and the projection theorem, Oxtoby's Measure and category for the difference-set theorem, Edgar's Integral, probability and fractal measure (1998) for compact subsets of analytic sets, the cell decomposition theorem of real algebraic geometry, and van den Dries's Lemma 4.1 cited above.

Bears on

• Problem 1154, which asks whether every $\alpha\in[0,1]$ is the Hausdorff dimension of some ring or field in $\mathbb R$: the Theorem shows that no real closed subfield of $\mathbb R$ that is an analytic set, in particular a Borel set, has dimension strictly between $0$ and $1$. The problem does not restrict the ring or field to analytic sets or to real closed fields, so the Theorem does not answer it.