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Statement

CH dimension is the Cartesian--Hausdorff dimension of Theorem 1's page: the limit of $(1/n)\dim(E^n)$, which is $0$ exactly when every finite Cartesian power of $E$ has Hausdorff dimension $0$.

Theorem 2 (§ 2, p. 4). "Let $E\subseteq\mathbb C$ be a subring and a Borel set. Then $E$ has zero CH dimension or $E=\mathbb R$ or $E=\mathbb C$."

Analytic sets (Remarks, p. 7). For an analytic set $X\subseteq\mathbb C$ the generated ring $\mathbb Z[X]$ is analytic, and if $\dim X^k>0$ for some $k$ then $\mathbb Z[X]$ is $\mathbb R$ or $\mathbb C$, according as $X\subseteq\mathbb R$ or not. The abstract (p. 1) states the consequence that an analytic subring of $\mathbb C$ of positive Hausdorff dimension is $\mathbb R$ or $\mathbb C$.

Source. 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. Theorem 2 on p. 4, Lemmas 2.1--2.4 on pp. 4--7 with their proofs, the Remarks on p. 7. Pages and labels are those of the authors' nine-page preprint identified on the source card; the journal edition was not compared.

Read depth. Claims checked: the theorem, the statements of Lemmas 2.1--2.4 and the Remarks were read clause by clause on the page images. The proofs were read for structure only; nothing here is independently reviewed.

Proof pointer

Section 2, pp. 4--7, following the proof of Theorem 1 with complex-linear functionals. Lemma 2.1 (pp. 4--6), the case of the complex Projection Theorem with one-complex-dimensional range, which the paper proves since it did not find it in print: a Borel $A\subseteq\mathbb C^k$ with $\dim A>2$ has image of positive two-dimensional Lebesgue measure under almost every $\mathbb C$-linear functional $\mathbb C^k\to\mathbb C$. The proof follows Mattila's argument for $\mathbb R$, through a measure of finite $2$-energy on $A$ and a bound on the surface measure of bands of the unit sphere. Lemma 2.2 (p. 6) gives $\varphi(E^k)=\mathbb C$ for a Borel additive subgroup of nonzero CH dimension, by Steinhaus's theorem in the plane. Lemma 2.3 (p. 6) makes $\varphi$ bijective on $E^k$ for a subring, as in Lemma 1.3. Lemma 2.4 (pp. 6--7): if a $\mathbb C$-linear functional maps $E^k$ bijectively onto $\mathbb C$ for a Borel additive subgroup $E$, then $k=1$ and $E=\mathbb C$, or $k=2$ and $E=\mathbb R$; here the first coordinate of the inverse is a continuous additive, hence $\mathbb R$-linear, map $\mathbb C\to\mathbb C$, and the cases follow from the possible real dimensions of its null space.

Dependencies

The proof of Lemma 2.1 imitates Mattila, Geometry of sets and measures in euclidean spaces, Thm. 9.7 and Lemma 3.11, and uses Edgar, Integral, probability and fractal measure, (3.2.7), for the finite-energy measure. Lemma 2.2 uses the planar Steinhaus theorem, credited to Ruziewicz; Lemma 2.4 uses the automatic continuity of Borel measurable homomorphisms (Banach, Théorie des opérations linéaires, Ch. I, Thm. 4; Kechris, Classical descriptive set theory, 9.10).

Bears on

No problem in the corpus. The real case, which bears on Problem 1154, is Theorem 1.