Improved packing of hypersurfaces in $\mathbb R^d$

Tongou Yang; Xianghong Chen; Yue Zhong

REVIEW 2 major objections 4 minor 29 references

Improved packing of hypersurfaces in $\mathbb R^d$

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper constructs a compact set in $\mathbb{R}^{d+1}$ containing a $d$-sphere of every radius in $[1,2]$ whose $\delta$-neighbourhood has measure $\lesssim_d |\log\delta|^{-2/d}$; for $d=2$ this order is optimal.

desk verdict Right idea, wrong exponent: the proof as typeset uses 2^{Md} where it needs 2^{M^d}, and that error breaks the central estimate for d≥2. read the letter →

arxiv 2501.03532 v1 pith:M665XTVJ submitted 2025-01-07 math.CA

Xianghong Chen , Tongou Yang , Yue Zhong This is my paper · ORCID

classification math.CA MSC 28A7542B25

keywords Kakeya-typesetsspherepackinghypersurfaceδ-neighbourhoodmeasureGaussiancurvaturegrid-pathestimatetangentcompression

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that for every dimension $d \ge 1$, one can place a $d$-sphere of every radius between 1 and 2 into a compact subset of $\mathbb{R}^{d+1}$ whose $\delta$-neighbourhood has Lebesgue measure at most a constant (depending only on $d$) times $|\log \delta|^{-2/d}$. For $d=2$ this order is the best possible, since a known lower bound forces $|N_\delta(K)| \gtrsim |\log \delta|^{-1}$. The construction works by a dyadic compression: curves are translated so that representative graphs become tangent at successive grid points, and the whole measure estimate is reduced to an elementary inequality for paths on a grid. The same mechanism applies to any Hölder-continuous one-parameter family of $C^{2,\alpha}$ hypersurfaces with Gaussian curvature bounded away from zero, yielding $|\log \delta|^{-(1+\alpha)/d}$.

What carries the argument

The engine is an iterative tangent-compression scheme. The parameter interval is split into $2^{Md}$ pieces, and the corresponding graph pieces are translated so that their representative functions match both value and gradient at successive points $x_j$ along a back-and-forth Hamiltonian path (visiting every interior grid point exactly once) through the interior points of a $J^d$ grid. At step $j$ the translation vectors have size about $\delta_0 2^{j-Md}$, and the total accumulated translation stays tiny because each piece is only moved when its binary index says so. The measure bound is reduced to the grid-path estimate (2.22), proved through Lemma 2.3: along any unit-step path $\{n_i\}$ in $\{1,\dots,J\}^d$ one has $\sum_{i=1}^j 2^i |n_j-n_i|^{1+\alpha} \lesssim 2^j M^{-1-\alpha}$, and this inequality is what removes the logarithmic loss.

What would settle it

For a fixed dimension $d\ge3$ and a small exponent $\alpha$ (say $\alpha=0.1$), compute the left side of (2.22) numerically for the back-and-forth path produced by the paper's induction, at $x=x_j$ and over a range of $M$ and $j$. If the ratio to $2^j M^{-1-\alpha}$ grows without bound, the key estimate (2.22) is false. A simpler check is to search exhaustively for a unit-step path in $\{1,\dots,J\}^d$ that violates Lemma 2.3.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the previously known upper bound for packing spheres, which carried an extra $\log |\log \delta|$ factor, can be sharpened to $|N_\delta(K)| \lesssim_d |\log \delta|^{-2/d}$. The proof establishes more: for any regular family of curved hypersurfaces $S_a$, defined by $\Phi(a,x)=0$ with Gaussian curvature uniformly bounded away from zero and $C^{2,\alpha}$ regularity, the same compression gives $|N_\delta(K)| \lesssim |\log \delta|^{-(1+\alpha)/d}$. The sphere case follows because the family $\Phi(a,x)=|x|-a$, $1 \le a \le 2$, satisfies these hypotheses with $\alpha=1$.

Load-bearing premise

The measure bound rests on the grid-path inequality (2.22), whose proof in Lemma 2.3 is only sketched: the summation-by-parts step is asserted, and the construction of the back-and-forth Hamiltonian path for general dimension is stated by induction. If that inequality failed for some $d$ or some $\alpha$, Proposition 2.1 and hence the sphere-packing theorem would not follow.

Editorial extensions

If this is right

For $d=2$, the new construction meets the lower bound $|N_\delta(K)| \gtrsim |\log \delta|^{-1}$, so the exact order of the $\delta$-neighbourhood measure for packing all circles of radii in $[1,2]$ is now known.
The $\delta$-level construction passes through a standard iterative argument to give a compact set of Lebesgue measure zero that still contains a $d$-sphere of every radius in $[1,2]$.
The same algorithm proves a general packing bound $|\log \delta|^{-(1+\alpha)/d}$ for Hölder-continuous one-parameter families of $C^{2,\alpha}$ hypersurfaces with Gaussian curvature bounded away from zero, including hyperbolic paraboloids.
The result holds in every dimension $d \ge 1$ and removes the $(\log|\log\delta|)^{2/d}$ factor from the previous sphere-packing construction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The grid-path inequality is the only step whose proof is sketched rather than fully written; a natural test is whether the same inequality holds for the specific paths used when $d \ge 3$, since the paper constructs those paths by induction without giving the explicit ordering.
The paper's remark that cylinders pack better than spheres suggests that the packing rate is governed by the dimension of the set of normal directions of the family; one could test this by computing sharp exponents for intermediate families that interpolate between cylinders and spheres.
For $d=1$, where optimality of $|\log \delta|^{-2}$ remains open, the same tangent-compression scheme might be inverted to build a lower-bound example, because the grid-path mechanism does not obviously favour upper bounds over lower bounds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know before you read this paper. First, the main result is likely right and is a genuine improvement: it removes the log log factor from Kolasa-Wolff and gets the sharp order in d=2, with a clean generalization to C^{2,α} hypersurfaces. Second, as typeset, the proof has a systematic exponent error: everywhere it writes 2^{Md}, it needs 2^{M^d}. That error is load-bearing.

In Proposition 2.1 the parameter interval is cut into 2^{Md} pieces, so each n has Md bits, but the grid of tangency points has about (M/2)^d points, which for d≥2 is far larger than Md. The binary grouping in Section 2.4 only makes sense for j ≤ Md; for the typical grid point with j > Md, the decomposition n = p + Σ ε_i 2^{i-1} collapses (p=0) and the thickness bound contains a factor 2^{j-Md} ≥ 1, destroying the M^{-1-α} fiber estimate. The same slip makes Proposition 3.1 claim a |log δ|^{-1-α} bound for spheres in d=2, contradicting the Kolasa-Wolff lower bound |N_δ| ≳ |log δ|^{-1}. The theorem's stated exponent (1+α)/d is what you get from 2^{M^d} scaling.

If you mentally replace M d with M^d everywhere, the proof is coherent and the arguments are largely standard: the implicit function theorem step, the Taylor expansions, and the grid-path estimate (2.22) are plausible. Lemma 2.3 is sketched but the summation-by-parts claim is believable. The deduction of the single compact set in the abstract from the per-δ theorem is asserted rather than written out, but that is a routine iteration.

The paper is worth engaging with after the typo is fixed. It builds on Yang-Zhong and extends it genuinely; the novelty is real, not a repackaging. The citation practice is fine. My recommendation: send it to a serious referee, but the referee should insist the M^d/M d confusion be resolved explicitly. It may be a LaTeX error, but as submitted the central estimate is not proved for d≥2.

Referee Report

2 major / 4 minor

Summary. The paper constructs, for each δ∈(0,1), a compact set Kδ in R^{d+1} that contains a translated copy of every d-sphere of radius between 1 and 2, with |N_δ(Kδ)| ≲_d |log δ|^{-2/d}. For d=2 this is the sharp order predicted by the Kolasa–Wolff lower bound. The proof proceeds by first establishing Proposition 2.1, a packing result for graphs of regular curved functions, via a discrete translation algorithm that compresses the graph family at a grid of tangency points. Proposition 3.1 then uses a partition of unity and the graph result to handle general curved hypersurfaces, and the sphere case follows as a corollary. The paper also states a generalisation to Hölder-continuous families of C^{2,α} hypersurfaces of nonzero Gaussian curvature.

Significance. If the proof is correct, the main result is a genuine improvement over the Kolasa–Wolff construction, removing a (log|log δ|)^{2/d} factor and establishing the sharp order of the δ-neighbourhood for d=2. The method is self-contained, uses only elementary Taylor expansions and a discrete combinatorial inequality, and is clearly presented. The generalisation to nonconvex curved hypersurfaces with nonzero Gaussian curvature is a useful extension of previous work. The paper also draws a helpful conceptual distinction between cinematic curvature and the curvature condition used here. The main risk is the scaling consistency of the construction, which is discussed in the major comments.

major comments (2)

[§2.1–2.4, in particular (2.2), (2.3), (2.16), (2.19)–(2.22)] The manuscript consistently types the exponent as 'M d', e.g. '2^{-M d}', 'M d/3', 'M d − m ≥ M d/3'. If this is read literally as M·d, the construction divides the parameter interval into 2^{M d} pieces, but the grid of tangency points has m = (M/2−1)^d + λ points. For d ≥ 2 and large M, m ≫ M d (e.g., d=2 gives m ≈ M^2/4 while M d = 2M). The binary decomposition (2.16) is then valid only for j ≤ M d; for j > M d the only possible p is 0, so the grouping is trivial and the thickness bound (2.19) cannot be proved for typical x′, whose closest grid point has index j > M d. Moreover, the line after (2.14), 'M d − m ≥ M d/3', is false under the literal reading for d ≥ 2. The intended notation is evidently M^d: this is consistent with the theorem exponent |log δ|^{-2/d}, with Proposition 3.1's final scale 2^{-N M_0^d} = 2^{-M^d}, and with the inequality M^d − m ≥ M^d/3. The proof therefore requires a systematic correction of every occurrence of 'M d' to 'M^d' in the exponents, and a re-verification of all estimates under that scaling. Without this correction, the measure bound (2.15) is not established and the claimed result would contradict the Kolasa–Wolff lower bound when d=2.
[§2.5, Lemma 2.3 and its remark] The key grid-path estimate (2.22) is reduced to Lemma 2.3, but the lemma is only proved for the exponent 2, and the remark asserts that the same bound holds for any exponent β by induction without giving the argument. Since the application requires β = 1+α with α ∈ (0,1], a complete proof is needed. This is a load-bearing estimate: if it fails, the bound (2.21) and hence the fiber measure estimate (2.15) do not follow. The claim is plausible, but as written it is a gap in the proof.

minor comments (4)

[Throughout, especially (2.7), (2.20), (2.21)] The repeated appearance of 'M d' instead of 'M^d' is more than a typographical nuisance: it makes the proof impossible to follow as typeset and obscures the central scaling. The authors should carefully correct all exponents and ensure that the notation is unambiguous.
[§2.4, after (2.20)] The text says 'we can replace ∂_a f(a_n, x_i) by ∂_a f(a_{⌊n⌋_i}, x_i)', but the expression in (2.20) is evaluated at x, not at x_i. The intended statement is that ∂_a f(a_n, x) is replaced by ∂_a f(a_{⌊n⌋_i}, x) using (1.3), and similarly for ∇_x f. This should be corrected to avoid confusing the reader.
[§1, abstract and Theorem 1.1] The abstract claims a single compact set K containing spheres of every radius in [1,2], while Theorem 1.1 produces a set K_δ for each δ. The passage from the theorem to the abstract is said to follow by a standard iterative argument, but no details are given. A brief explanation of that diagonal argument would improve readability.
[§2.1.1] The construction of the continuous path x̃_j for general d is described only by induction and an example. Since the path order is used in the key estimate (2.22), a precise recursive definition for all d would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's construction is self-contained, and its self-citation to [YZ24] is comparative rather than load-bearing.

full rationale

The derivation chain is self-contained. Proposition 2.1 constructs the set F_M directly from the given regular curved function f, using only the assumptions in Definition 1.2, Taylor expansion, the binary expansion of indices, and Lemma 2.3. Proposition 3.1 reduces the hypersurface case to Proposition 2.1 through a fixed number of coordinate charts and a partition of unity, with no fitted parameter or externally supplied input. Theorem 1.5 and Theorem 1.1 then follow by applying these propositions to the sphere family. The measure bound |log δ|^{-(1+α)/d} is an output estimated from the construction, not an input assumed in the hypotheses or in the cited work. The citation to the authors' own [YZ24] appears in the introduction as a comparison ('Theorem 1.3 generalises [YZ24, Theorem 1.3]') and the proof is written out rather than imported; no load-bearing step invokes [YZ24] as a black box. The sketched proof of Lemma 2.3 and the possible indexing mismatch between 2^{Md} and m are rigor or correctness concerns, not circularity, since neither reduces the target estimate to an equivalent statement of itself or to a fitted parameter.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new postulated entities. It relies on standard tools of real analysis and on the explicit regularity hypotheses of Definition 1.2.

assumptions (4)

standard math Implicit function theorem
Used to solve (2.6) for the translation u in Section 2.2.1, relying on the curvature condition (1.1).
standard math Taylor's theorem with remainder
Frequent Taylor expansions in Sections 2.3 and 2.4 with Hölder error terms.
standard math Existence of a Hamiltonian path on the grid graph (M/2-1)^d
Section 2.1.1 constructs a continuous path; for general d it is asserted by induction.
standard math Lebesgue measure subadditivity and basic measure estimates
Used in Section 2.4 to combine estimates over many intervals.

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Pith. "Pith review of Improved packing of hypersurfaces in $\mathbb R^d$." pith.science (2026). https://pith.science/paper/M665XTVJ

@misc{pith2026250103532,
  author       = {Pith},
  title        = {Pith review of: Improved packing of hypersurfaces in $\mathbb R^d$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M665XTVJ}},
  note         = {Machine review of arXiv:2501.03532}
}

abstract

For $d\ge 1$, we construct a compact subset $K\subseteq \mathbb {R}^{d+1}$ containing a $d$-sphere of every radius between $1$ and $2$, such that for every $\delta\in (0,1)$, the $\delta$-neighbourhood of $K$ has Lebesgue measure $\lesssim |\log \delta|^{-2/d}$. This is the smallest possible order when $d=2$, and improves a result of Kolasa-Wolff (Pacific J. Math., 190(1):111-154, 1999). Our construction also generalises to Holder-continuous families of $C^{2,\alpha}$ hypersurfaces with nonzero Gaussian curvature.

Figures

Figures reproduced from arXiv: 2501.03532 by the authors.

**Figure 1.** Translations of the annuli after 3 steps is constant on intervals I1 of the form (k2 −Md 0 ,(k + 1)2−Md 0 ), such that the 2 −Md 0 neighbourhood of the subset [ a∈I0 {x + w1(a) : x ∈ Sa} ∩ Γ1 (3.4) has Lebesgue measure ≲ M−1−α 0 . 3.3. Step 2. Now we fix such an interval I1 of length 2−Md 0 . After the first step of translations w1, the graph of x 7→ f2(a, x) becomes that of x 7→ f2(a, x+u)+v for some suitable u, v … view at source ↗

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

A. S. Besicovitch. On K akeya's problem and a similar one. Math. Z. , 27(1):312--320, 1928

work page 1928

[2] [2]

A. S. Besicovitch. The K akeya problem. Amer. Math. Monthly , 70:697--706, 1963

work page 1963

[3] [3]

A. S. Besicovitch and R. Rado. A plane set of measure zero containing circumferences of every radius. J. London Math. Soc. , 43:717--719, 1968

work page 1968

[4] [4]

Chang and M

A. Chang and M. Cs\"ornyei. The K akeya needle problem and the existence of B esicovitch and N ikodym sets for rectifiable sets. Proc. Lond. Math. Soc. (3) , 118(5):1084--1114, 2019

work page 2019

[5] [5]

M. Chen, S. Guo, and T. Yang. A multi-parameter cinematic curvature. 2023. arXiv:2306.01606

arXiv 2023

[6] [6]

C \' o rdoba

A. C \' o rdoba. The fat needle problem. Bull. London Math. Soc. , 25(1):81--82, 1993

work page 1993

[7] [7]

X. Chen, L. Yan, and Y. Zhong. On the generalized H ausdorff dimension of B esicovitch sets. 2023. arXiv:2304.03633

work page Pith review arXiv 2023

[8] [8]

R. O. Davies. Some remarks on the K akeya problem. Proc. Cambridge Philos. Soc. , 69:417--421, 1971

work page 1971