REVIEW 3 major objections 4 minor 19 references
Calculating Gromov-Hausdorff distance by means of asymptotic dimension
T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read The paper claims a general lower bound for Gromov–Hausdorff distance using asymptotic dimension and a scaling similarity, and proves that the distance between the integer lattice Z² and the Euclidean plane R² is exactly the Hausdorff…
desk verdict A genuine new trick: asymptotic dimension gives GH lower bounds, and the main examples are correct after a small limiting step the paper omits. 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 machinery is the r-disjoint cover definition of asymptotic dimension together with the stabilizer St_X = {λ > 0 : λX is isometric to X}. A correspondence with distortion < r maps k r-disjoint uniformly bounded families of A to ε-disjoint uniformly bounded families of X; multiplying by a stabilizer λ > 1 stretches both separations and diameters, so the ratio stays fixed but the scale grows without bound, contradicting asdim X ≥ n whenever k ≤ n. The chessboard coloring of Z² supplies the two bounded families used in the main example.
What would settle it
Construct a correspondence between Z² and R² with distortion strictly below √2; by Proposition 1 this would give d_GH(Z²,R²) < √2/2 and refute the paper's equality. A direct check of Example 1 at r = √2 is also decisive: the chessboard color classes have minimum same-color distance exactly √2, so under Definition 9 they are not strictly √2-disjoint, and the theorem must be applied along a sequence r → √2.
Extended reading notes
Core claim
Theorem 3 is the paper's central discovery: a lower bound on d_GH(A,X) obtained without building correspondences. For metric spaces X and A, if asdim X ≥ n and St_X ≠ {e}, and if A admits a cover by k r-disjoint families of uniformly bounded subsets with 1 ≤ k ≤ n, then d_GH(A,X) ≥ r/2. The proof takes an arbitrary correspondence R with dis R < r, pushes the r-disjoint families from A into X, where they become ε-disjoint uniformly bounded families, and then uses a stabilizer λ > 1 to scale them, producing arbitrarily large separated families that contradict asdim X ≥ n. For X = R² and A = Z², the two chessboard color classes play the role of the two families, yielding d_GH(Z²,R²) = d_H(Z²,R²) = √2/2.
Load-bearing premise
Everything rests on finding k ≤ n uniformly bounded families in A whose mutual distances are strictly greater than r, together with a genuine scaling similarity of X; if the separation is only equal to r, the positive margin used in the proof collapses.
Editorial extensions
If this is right
- For the integer lattice and the Euclidean plane, the exact value d_GH(Z²,R²) = √2/2 follows, matching the Hausdorff distance of the standard inclusion.
- For the comb-shaped set A = R×{0} ∪ ∪_{n∈Z} {n}×R inside R², the same theorem gives d_GH(A,R²) = d_H(A,R²) = 1/2.
- Any n-dimensional normed space has asymptotic dimension n, so the theorem applies to every such space with a non-trivial stabilizer, giving a broad class of exact lower bounds.
- When a space A admits k ≤ n uniformly bounded r-disjoint covering families, the lower bound d_GH(A,X) ≥ r/2 is automatic; thus any isometric embedding of A into X whose Hausdorff distance equals r/2 is automatically Gromov–Hausdorff optimal.
- The method converts exact Gromov–Hausdorff computation into a finite covering problem, potentially yielding new exact distances wherever such separated bounded covers are easy to describe.
Reading between the lines
- A natural extension, not pursued in the paper, is to apply the theorem to Zⁿ inside Rⁿ with the ℓ∞ metric: the parity coloring gives two uniformly bounded families separated by distance 1, so for every n ≥ 2 the same argument would yield d_GH(Zⁿ,Rⁿ) = 1/2, matching the Hausdorff bound.
- If the strict inequality in the definition of r-disjointness is relaxed to d ≥ r, a limiting version of Theorem 3 would likely hold, making Example 1 apply directly at r = √2 instead of requiring a sequence r → √2.
- The proof suggests a sharper principle: whenever an isometric embedding realizes the Hausdorff distance, it is Gromov–Hausdorff optimal exactly when the source space admits r-disjoint bounded covers with k ≤ asdim of the target.
- The asymmetric nature of the argument points toward a dual statement: covering properties of the target space might yield upper bounds on d_GH, complementing the lower bounds obtained here.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a method for deriving lower bounds on Gromov–Hausdorff distances using asymptotic dimension. The main result, Theorem 3, asserts that if X has asymptotic dimension at least n and nontrivial stabilizer St_X, and if A admits a cover by k ≤ n r-disjoint families of uniformly bounded subsets, then d_GH(A,X) ≥ r/2. The proof proceeds by contradiction: a correspondence with distortion less than r would transfer the r-disjoint bounded families of A to ε-disjoint bounded families covering X, and iterating a similarity of X would produce arbitrarily separated bounded covers, contradicting asdim X ≥ n. The paper applies this to show d_GH(Z^2,R^2) = √2/2 using a chessboard coloring of Z^2, and to compute d_GH(A,R^2) = 1/2 for a comb-like subset A of R^2.
Significance. The idea of using asymptotic dimension as an obstruction to small Gromov–Hausdorff distance is original and potentially valuable for unbounded spaces with self-similarities. The equality d_GH(Z^2,R^2) = √2/2 is a clean, nontrivial example, and the method avoids constructing optimal correspondences explicitly. The paper is transparent about relying on standard results (asdim R^n = n, the correspondence formula for d_GH). However, the proof as written has several load-bearing gaps: the scaling operation on subsets of a general metric space is not formally defined, and the flagship application invokes r-disjointness at a value that violates the strict inequality in Definition 9. These issues are fixable, but they currently make the paper incomplete.
major comments (3)
- [Section 3, proof of Theorem 3] The notation λ^n V^i is not defined. St_X is defined as the set of λ for which λX is isometric to X, which implies the existence of a bijection f: X → X with d(f(x),f(y)) = λ^{-1} d(x,y) (or its inverse providing scale λ), but the proof never introduces such a similarity. The sentence 'Consider the families λ^n V^1, ..., λ^n V^k of subsets of X' is therefore meaningless for a general metric space. This is load-bearing because the contradiction relies on producing arbitrarily separated bounded covers of X. The proof must explicitly fix a similarity f of X with scale μ > 1 and define μ^n V = f^n(V), or otherwise specify how St_X acts on subsets.
- [Example 1] The chessboard color classes are not √2-disjoint under Definition 9, which requires d(U_α,U_β) > r. For distinct same-color points, the distance can be exactly √2, e.g., between (0,0) and (1,1). Therefore Theorem 3 cannot be applied with r = √2. The intended bound, d_GH(Z^2,R^2) ≥ √2/2, can be recovered by applying Theorem 3 for every r < √2 and taking the supremum as r → √2, but this limiting argument is absent from the text. This is a genuine error in the paper's main example and must be corrected.
- [Example 2] The assertion that A admits a cover by two families of uniformly bounded subsets that are 1-disjoint is stated without proof. With the strict inequality in Definition 9, the existence of such a cover is not immediate: any horizontal segment of A has distance 0 from the adjacent vertical lines, so the assignment of pieces to the two families must be described explicitly (or, if the figure is meant to illustrate a limiting r < 1 construction, that should be stated). Please provide the explicit families or a precise description of the cover used in Figure 2, and verify that the strict r-disjointness condition is satisfied.
minor comments (4)
- [Abstract and Introduction] There are several typographical errors and missing spaces, e.g., 'Inthispaper', 'theGromov–Hausdorff', and 'dGH (R2, Z2) = dH (R2, Z2)' in the abstract. The text should be cleaned up.
- [Definition 7] The stabilizer St_X is defined in terms of λX being isometric to X, but the proof of Theorem 3 treats St_X as though it acts on X by similarities. A sentence explaining the action of St_X on subsets would resolve the ambiguity.
- [Definition 10] The definition of r-disjoint families uses a strict inequality d(U_α,U_β) > r. This is compatible with the proof of Theorem 3, but the strictness is easy to overlook; the paper should perhaps remind the reader of this convention when applying the definition in examples.
- [References] Reference [3] contains a typo: 'Guolinag Yu' should be 'Guoliang Yu'.
Circularity Check
No circularity found: Theorem 3 is a deductive implication from stated hypotheses, and the chessboard application uses standard external facts rather than fitting the target distance.
full rationale
The derivation chain is self-contained in the relevant sense. Theorem 3 (Section 3) is a direct implication: from r-disjoint uniformly bounded covers of A, a nontrivial stabilizer of X, and asdim X ≥ n, it constructs a contradiction with any correspondence having disR < r. The lower bound d_GH(Z^2,R^2) ≥ √2/2 then instantiates this theorem with the two chessboard families, and the upper bound is the standard Hausdorff inequality. No parameter is fitted to the target value and no conclusion is assumed as an input. The cited supports (Proposition 1, Theorem 1, the stabilizer definition from [4]) are external standard results, not self-citations by this author. The only noteworthy defect is Example 1 (Section 4), which claims the color classes are '√2-disjoint': Definition 9 requires strict inequality, while same-color points can be exactly √2 apart, so the families are r-disjoint only for r < √2; the proof needs the usual limiting argument r → √2. That is a correctness gap, not circularity, because it concerns the applicability of the theorem rather than the theorem reducing to its own conclusion.
Assumptions & free parameters
assumptions (5)
- standard math Asymptotic dimension of R^n is n (Theorem 1, from Nowak-Yu).
- standard math Equivalent definition of asymptotic dimension via r-disjoint families (Definition 10, from Bell-Dranishnikov).
- standard math Formula 2 d_GH(X,Y) = inf disR over correspondences (Proposition 1, from Burago-Burago-Ivanov).
- standard math Every two norms on a finite-dimensional space are equivalent, so asdim of any normed space of dimension n is n (Theorem 2).
- domain assumption The stabilizer St_X = {lambda > 0 : lambda X is isometric to X} is a multiplicative subgroup of R_{>0} (stated after Definition 7).
Cite this review
Pith. "Pith review of Calculating Gromov-Hausdorff distance by means of asymptotic dimension." pith.science (2026). https://pith.science/paper/VXF4UX3P
@misc{pith2026250518158,
author = {Pith},
title = {Pith review of: Calculating Gromov-Hausdorff distance by means of asymptotic dimension},
year = {2026},
howpublished = {\url{https://pith.science/paper/VXF4UX3P}},
note = {Machine review of arXiv:2505.18158}
}
abstract
In this paper, we apply the concept of asymptotic dimension to calculating Gromov-Hausdorff distances between some unbounded metric spaces. For example, we show that the Gromov--Hausdorff between $\mathbb{R}^2$ with the Euclidean metric and $\mathbb{Z}^2$ equals the Hausdorff distance between them: $d_{GH}(\mathbb{R}^2, \mathbb{Z}^2) = d_H(\mathbb{R}^2, \mathbb{Z}^2)$.
Reference graph
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Reviewed August 16, 2026 · model on record in the stance chip above.
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