REVIEW 7 minor 4 references
An uncountable subring of $\mathbb R$ with Hausdorff dimension zero
T0 review · 0 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read An uncountable subring of $\mathbb{R}$ with Hausdorff dimension zero
desk verdict Explicit, choice-free Borel subring of R with Hausdorff dimension zero; the construction is sound and the two soft spots are minor and repairable. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the sparse digit-position set $S=\{0\}\cup\{2^m:m\ge0\}$ together with its iterated sumsets $nS$. Elements of $A_n$ are series $\sum_{k\in nS} x_k 2^{-k}$ with integer coefficients bounded by some $t$. The sparseness of $nS$ drives the dimension computation: the number of allowed positions up to $\ell$ is at most $(2+\log_2\ell)^n$, so the natural truncation covers shrink fast enough to force the $s$-dimensional Hausdorff measure to vanish for every $s>0$. Multiplicative closure is carried by Lemma 2, a uniform bound on the ordered representation count $c_n(k)$, the number of ways to write $k$ as a sum of $n$ elements of $S$. That bound transfers a product of two bounded-digit series into another series of the same type with bounded digits, placing the product in $A_{m+n}$.
What would settle it
Search for integers k whose number of ordered representations as a sum of n elements of S is unbounded as k grows; any unbounded sequence would falsify Lemma 2 and break the multiplication-closure argument. More directly, exhibit two elements of A whose product, expanded at positions in any (m+n)S, has coefficient sequence not bounded by any fixed integer.
Extended reading notes
Core claim
Let $S=\{0\}\cup\{2^m:m\ge 0\}$ and let $nS$ be the set of all sums of $n$ elements of $S$. The paper defines $A_n$ as the set of numbers of the form $\sum_{k\in nS} x_k 2^{-k}$ with integer coefficients bounded in absolute value by some $t$. It proves that each $A_n$ is an uncountable subgroup of $\mathbb{R}$ (Theorem 1) and that $A_n$ has Hausdorff dimension zero (Theorem 2). Letting $A=\bigcup_{n\ge 1} A_n$, the paper proves $A$ is closed under multiplication (Theorem 3) using a uniform bound on the number of representations of an integer as a sum of $n$ elements of $S$ (Lemma 2). Hence $A$ is an uncountable proper Borel subring of $\mathbb{R}$ of Hausdorff dimension zero; Corollary 1 shows its rational elements are exactly the dyadic rationals, so $A$ is not a field.
Load-bearing premise
The proof that products stay inside the ring depends on the claim that for each n there is a uniform upper bound, independent of the integer k, on the number of ways to write k as a sum of n numbers from {0,1,2,4,8,...}; if that bound failed, product coefficients could grow without bound and the ring could spill outside the construction.
Editorial extensions
If this is right
- There is an explicitly defined uncountable proper Borel subring of $\mathbb{R}$ with Hausdorff dimension zero, so the phenomenon of large algebraic structures inside geometrically negligible sets is not limited to additive subgroups.
- Every rational element of $A$ has the form $a/2^k$, so $A$ contains no rational like $1/3$ and is therefore not a field.
- Each element of $A$ has binary expansions containing arbitrarily long runs of equal digits (Proposition 1), a concrete Diophantine property that can be studied independently.
- The construction goes through for any digit set $T\cup\{0\}$ with bounded logarithmic density (Remark 1), yielding a whole family of zero-dimensional subrings rather than a single example.
- The ring $A$ is a nested union of uncountable additive subgroups $A_n$ of dimension zero, so the step from subgroups to a subring is exactly where the uniform representation-count bound enters.
Reading between the lines
- The paper does not pursue this, but the same construction with $S$ replaced by powers of any integer $b\ge3$, or by other sparse sets satisfying the logarithmic-density condition, should produce further explicit zero-dimensional subrings.
- Because Corollary 1 excludes non-dyadic rationals, the method cannot directly produce a subfield; an uncountable Borel subfield of $\mathbb{R}$ of dimension zero, if one exists, would need a different digit set or extra closure machinery.
- The explicit covers suggest a computational check: for fixed $n$ and $t$, one can enumerate the truncated expansions and numerically verify that the predicted $\delta$-covers shrink as claimed, giving an independent confirmation of the dimension-zero bound in small cases.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs an explicit set A = ∪_{n≥1} A_n, where A_n consists of real numbers representable by a base-2-type series with bounded integer coefficients supported on the set nS, the n-fold sumset of S = {0,1,2,4,8,...}. The authors prove that each A_n is an uncountable subgroup of R, that each truncated stage A_{n,t} has Hausdorff dimension zero by an explicit interval-covering argument, and hence that A has Hausdorff dimension zero. They then prove that A is closed under multiplication using a uniform bound on the number of representations of an integer as a sum of n elements of S (Lemma 2), making A a subring. A final argument shows that the only rational numbers in A are the dyadic rationals, so A is a proper subring and is not a field, and the paper also asserts that A is Borel. The main claim of the manuscript is that A is an uncountable proper subring of R with Hausdorff dimension zero.
Significance. If the small gaps noted below are repaired, this is a short, explicit, parameter-free construction of an uncountable proper subring of R with Hausdorff dimension zero, complementing the known results of Erdős–Volkmann (subgroups of any Hausdorff dimension), Davies (CH-dependent non-Borel subrings), and Edgar–Miller (Borel subrings have dimension zero). The proof is self-contained and verifies the ring and dimension properties directly from the definitions, with Lemma 2 as the key technical estimate. The construction uses no free parameters and no choice principle, and the main claims are falsifiable in the sense that each step is checked against the definitions of Hausdorff measure, subgroups, and rings. Once the Borel assertion is supplied with a proof, the paper also provides an explicit Borel example, which is a useful complement to the non-Borel examples in the literature.
minor comments (7)
- [Lemma 2] The induction step uses c_{n-1}(k-2^m) even when k-2^m = 0, although c_{n-1} is defined only for positive integers in the lemma statement. This is a small but real gap in a lemma that underpins Theorem 3; it should be repaired, for example by defining c_j(0) = 1 (the all-zero representation) and requiring b_j ≥ 1, so that the estimate c_n(k) ≤ n b_{n-1}(1 + log_2 n) remains valid.
- [Section 2, Borel claim] The opening of Section 2 states that the constructed ring is a Borel set, but no proof is given. This follows because each A_{n,t} is the continuous image of the compact product [-t,t]^{nS} and hence compact, so A is sigma-compact; the authors should include this argument or explicitly mark the Borel property as a separate claim.
- [Theorem 1] The uncountability argument invokes uniqueness of binary expansions, but the sums are supported on the sparse set nS rather than on all of N0. The claim is true (at the first differing position, the tail over nS is strictly smaller than the digit gap because nS omits infinitely many integer positions), but a one-sentence proof would make the argument fully rigorous.
- [Theorem 3] In the proof of Theorem 3, the sentence 'for every k ∈ nS there exist x_k ∈ [-t,t]_Z and y_k ∈ [-t',t']_Z' should refer to k ∈ mS for the x_k coefficients; as written the index set is inconsistent with the displayed sums.
- [Theorem 2, proof of Eq. (2)] The line 'ℓ ≥ 2c_l^r − 2 where r = 1/n' appears to be a typographical corruption of ℓ ≥ 2^{c_l^{1/n} - 2}, which is what the preceding inequality c_l ≤ (2 + log_2 ℓ)^n actually yields; please correct the exponent.
- [Corollary 1] The proof only establishes that a rational element of A has a finite binary expansion; the converse, that every dyadic rational lies in A, should be stated (it follows from 1/2^k ∈ A_1 for k ∈ S and closure under addition).
- [Throughout] There are several typographical errors: 'A is a a subring' in Theorem 3, the inconsistent spelling of Erdős, and 'for every z ∈ N' in Lemma 1 should be 'for every n ∈ N'.
Circularity Check
No circularity: the construction and proofs are self-contained, and cited works are used only as context or standard measure-theory facts.
full rationale
The paper explicitly constructs the sets A_n and A from the digit set S, then proves subgroup closure, Hausdorff dimension zero, and multiplication closure directly from the definitions. The Hausdorff dimension argument is a self-contained covering estimate using Lemma 1, a counting bound derived within the paper. The ring closure relies on Lemma 2, a uniform bound on ordered representations of k as sums of n elements of S; this is proved by an induction that is internal to the paper. No parameter is fitted, no prediction is made from a subset of data, and no prior result by the authors is invoked as a load-bearing premise. The cited references [1], [2], and [4] are used for background (e.g., existence of Borel subgroups of every Hausdorff dimension and standard facts about Hausdorff measure) rather than to establish any step in this construction. The only identified issue is a minor technical gap in Lemma 2: when k - 2^m = 0, the induction invokes c_{n-1}(0), which is not defined by the lemma's statement; however, this is an ordinary proof gap, not circularity, and it is immediately repairable by setting c_{n-1}(0) = 1. Because the central claim is verified by direct construction and estimation, the circularity score is 0.
Assumptions & free parameters
assumptions (2)
- standard math Standard properties of Hausdorff measure and dimension, including countable subadditivity and the countable union formula dim_H(union C) = sup dim_H(A) over A in C, as stated in Section 1 and cited to Falconer [4].
- standard math Basic arithmetic of powers of two and counting of integers in logarithmic intervals, used in Lemma 1 and Lemma 2.
Cite this review
Pith. "Pith review of An uncountable subring of $\mathbb R$ with Hausdorff dimension zero." pith.science (2026). https://pith.science/paper/IQG52QMQ
@misc{pith2026241113519,
author = {Pith},
title = {Pith review of: An uncountable subring of $\mathbb R$ with Hausdorff dimension zero},
year = {2026},
howpublished = {\url{https://pith.science/paper/IQG52QMQ}},
note = {Machine review of arXiv:2411.13519}
}
read the original abstract
We construct a subring as mentioned in the title (hence this subring has Lebesgue measure zero).
Reference graph
Works this paper leans on
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[1]
G. A. Edgar and C. Miller, Borel subrings of the reals, Proc. Amer. Math. Soc.131(4) (2002), 1121–1129
work page 2002
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[2]
P. Erd˝ os and B. Volkmann, Additive Gruppen mit vorgegebener Hausdorffscher Dimension,J. Reine Angew. Math221 (1966), 203–208
work page 1966
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[4]
K. Falconer, Fractal geometry, Mathematical foundations and applications(Second Edition), John Wiley & Sons Inc., Hoboken, NJ, 2003. Stephan Baier, Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute, PO Belur Math, Howrah, West Bengal 711202, India Email address: stephanbaier2017@gmail.com Shameek Paul , Departme...
work page 2003
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[3]
Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), 206–212
K. Falconer, On the Hausdorff dimensions of distance sets, Mathematika 32 (1985), 206–212
work page 1985
Reviewed August 12, 2026 · model on record in the stance chip above.
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