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Statement

Setting (§ 3, p. 7). $p$ is a prime, $\mathbb Q_p$ the field of $p$-adic numbers with its ultrametric absolute value, and $\mathbb Z_p$ the compact subring of $p$-adic integers. Haar measure $\lambda$ is normalized by $\lambda(\mathbb Z_p)=1$; $\mathbb Z_p$ is the union of $p$ translates of $p\mathbb Z_p$, so its Hausdorff dimension is $1$, and one-dimensional Hausdorff measure is Haar measure. CH dimension is the Cartesian--Hausdorff dimension of Theorem 1's page.

Theorem 3 (p. 7). "Let $E\subseteq\mathbb Q_p$ be a subring and a Borel set. Then $E$ has zero CH dimension or $E=\mathbb Q_p$ or $E=\mathbb Z_p$."

Finite extensions (p. 8). The paper adds, with the proofs described as similar again, that if $K$ is a finite algebraic extension field of $\mathbb Q_p$ and $E\subseteq K$ is a subring and a Borel set, then $E$ has zero CH dimension or is a closed subring.

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 3 on p. 7, Lemmas 3.1--3.4 on pp. 7--8, the remark on finite extensions on p. 8. 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 3.1--3.4 and the remark on finite extensions were read clause by clause on the page images. The paper gives no full proof; nothing here is independently reviewed.

Proof pointer

Section 3, pp. 7--8. The paper says the proof is essentially that of Theorem 1, gives remarks on the differences, and leaves the details to the reader. The lemmas it states run parallel to Lemmas 1.1--1.4: Lemma 3.1, a Borel $A\subseteq\mathbb Q_p^k$ with $\dim A>1$ has image of positive Haar measure under almost every linear functional $\mathbb Q_p^k\to\mathbb Q_p$, for the max norm on $\mathbb Q_p^k$ and the product measure $\lambda^k$; Lemma 3.2, a Borel additive subgroup of nonzero CH dimension has some $\varphi(E^k)$ an open subgroup; Lemma 3.3, for a subring $\varphi$ can be taken to map $E^k$ bijectively onto an open subgroup; Lemma 3.4, a Borel additive subgroup on whose $k$-th power a linear functional is a bijection onto an open subgroup has $k=1$ and is itself an open subgroup. The open subgroups of $\mathbb Q_p$ are $\mathbb Q_p$ and the $p^n\mathbb Z_p$ (p. 7).

Dependencies

Versions of the Dimension Inequality (Edgar, Integral, probability and fractal measure, (3.2.12)) and of Steinhaus's theorem (Hewitt and Ross, Abstract harmonic analysis I, Cor. 20.17) valid for $\mathbb Q_p$, and automatic continuity of Borel measurable homomorphisms of complete metric groups (Banach; Kechris, 9.10; Topsøe and Hoffmann-Jørgensen, 2.3.1).

Bears on

No problem in the corpus.