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Space-filling surfaces: sharp H\"older continuous parameterizations from squares to cubes

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Pith's one-line read This paper constructs surjective maps from squares to cubes that are (2/3)-Hölder continuous, settling the attainability half of Arnold's problem 1988-5.

desk verdict A complete, constructive proof of the critical Hölder exponent for maps from m-cubes to (m+1)-cubes; the only real weakness is the unproved use of Stong's lattice bijection, which is a published theorem. read the letter →

arxiv 2608.21246 v1 pith:OYYV3B7J submitted 2026-08-21 math.MG math.CA

Matthew Badger , Kevin Palmer This is my paper · ORCID
classification math.MGmath.CA MSC 26B3528A8051F3054C0511A63
keywords space-fillingsurfacesHöldercontinuousparameterizationsArnold'sproblem1988-5criticalexponentself-similarsetsStonglatticebijectionsPeanocurvesM-adicrepresentations
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

This paper proves that for every dimension $m\geq 2$ there is a surjective map from the $m$-dimensional unit cube onto the $(m+1)$-dimensional unit cube whose Hölder exponent is $m/(m+1)$, the largest exponent allowed by a volume-counting argument. In particular, it constructs $(2/3)$-Hölder continuous maps from the square onto the cube. Arnold's problem 1988-5 asked for the critical exponent and whether it is attained; the critical value was already known, and this paper settles the attainability half with an explicit construction and an explicit Hölder constant. The construction follows a hint recorded in [Sem03], combining lattice bijections from [Sto98] with self-similar fractals.

What carries the argument

The load-bearing object is an X-shaped self-similar set $E_\infty\subset\mathbb{R}^2$ with similarity dimension $(m+1)/m$, together with a carefully constructed $\alpha$-Hölder surjection $g:[0,1]\to E_\infty$ with constant at most $4$. The proof uses graph-directed iterated function systems to keep this constant small. The final map is essentially the composition $[0,1]^m \to E_\infty^m \to L(E_\infty^m) \to [0,1]^{m+1}$, where $L$ is a linear map that, on a digit-restricted lattice set $X_m^m$, gives a bijection onto $\mathbb{Z}^{m+1}$; this lattice bijection is what lets the construction place lattice points inside $L(E_\infty^m)$ at every scale.

What would settle it

Check Stong's map explicitly on a finite box: take all points of $X_m^m$ with coordinates in $[-M^N,M^N]$ and compare their images under $F$ with the corresponding box in $\mathbb{Z}^{m+1}$; if any lattice point is missed or any collision occurs for a large $N$, the construction's load-bearing bijection is false. Alternatively, for $m=2$, sample the final map $f$ on a fine grid of the square and test whether every point of a fine grid of the cube is hit within the discretization error; a persistent uncovered region would refute surjectivity.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for all $m\geq 2$ there exists a surjection $f:[0,1]^m\to[0,1]^{m+1}$ satisfying the Hölder bound $\|f(x)-f(y)\|_\infty \leq 2^{m+2}(2^m-1)\|x-y\|_\infty^{m/(m+1)}$ for all $x,y$. The proof realizes $f$ as a composition of an $m$-fold product of a Hölder curve $g$ filling an X-shaped self-similar set $E_\infty$, a linear map $L$, and a projection onto the cube. Because the exponent is the best possible, the critical exponent from $m$-cubes to $(m+1)$-cubes is attained, answering the second part of Arnold's problem.

Load-bearing premise

The proof quotes Stong's theorem from [Sto98] that the map from the digit-restricted lattice set $X_m^m$ onto $\mathbb{Z}^{m+1}$ is a bijection; if that bijection failed, the scaling argument that places lattice points inside the image would break and surjectivity of the final map would no longer follow.

Editorial extensions

If this is right

  • If the theorem is correct, the critical Hölder exponent $m/n$ is attained for all $1\leq m\leq n$, by composing the maps between consecutive dimensions.
  • The same construction gives $(m/n)$-Hölder surjections from $\mathbb{R}^m$ onto $\mathbb{R}^n$ by tiling with separated cubes and extending via McShane's extension theorem.
  • For compact convex targets, there are $(m/n)$-Hölder surjections from any convex set with interior onto any compact convex set.
  • The explicit Hölder constant $2^{m+2}(2^m-1)$ is finite for every $m$, and the lower bound $2^{m/(m+1)}$ shows the constant cannot be smaller than this value.
  • The paper leaves open whether there is a universal Hölder constant independent of $m$; its construction gives an upper bound that grows exponentially in $m$.

Reading between the lines

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

  • Editor's inference: the same lattice-to-fractal transfer might produce sharp parameterizations for other pairs of dimensions $(m,n)$, especially in the range $m<n<2m$ where grid-based constructions do not work; the present proof only treats $n=m+1$.
  • Editor's inference: the explicit small constant for $g$ may make the construction usable in quantitative measure-geometry settings, since a large implicit constant would make estimates impractical.
  • Editor's inference: numerical exploration for $m=2$ could estimate how far the true optimal constant is from the lower bound $2^{2/3}$ and from the paper's upper bound $48$; the wide gap suggests the universal-constant question may have a positive answer.
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Editorial analysis

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

Referee Report

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Summary. The paper constructs, for every integer m≥2, an m/(m+1)-Hölder continuous surjection f:[0,1]^m → [0,1]^{m+1} with explicit constant 2^{m+2}(2^m−1), thereby attaining the critical Hölder exponent established by Shchepin and settling the attainability part of Arnold's problem 1988-5. The construction combines a self-contained parameterization g:[0,1]→E∞ of a self-similar 'X-fractal' with Hölder constant 4 (Section 2) with Stong's bijection F:X_m^m→Z^{m+1} between integer lattices (Proposition 3.1). A bi-Lipschitz estimate (Proposition 3.2) and a lemma on lattice points (Lemma 3.5) show that the linear image of the product fractal contains a cube, which is then projected and affinely scaled onto the target cube. Corollaries extend the result to arbitrary dimensions and convex bodies.

Significance. If the main theorem holds, it resolves a long-standing question of Arnold on the optimal Hölder regularity of maps between cubes of consecutive dimensions. The proof is constructive and fully explicit, with no fitted parameters; the estimates in Section 2 are detailed and self-contained, and the use of Stong's bijection is an external but published result. The paper also gives a lower bound for the Hölder constant (Lemma 4.3) and several corollaries for Euclidean spaces and convex bodies. The work is a significant contribution to the theory of space-filling curves and surfaces and to geometric measure theory.

minor comments (7)
  1. [Lemma 3.4] The statement E∞ = ∪_{k≥1} γ_k(S_k) is false; the proof only demonstrates that the union is dense in E∞. Indeed, the union is countable, whereas E∞ is an uncountable self-similar set of positive dimension. The main theorem does not rely on the false equality—Lemma 3.5 uses only the inclusion γ_k(S_k)⊂E∞—but the statement should be corrected to assert that the closure of the union equals E∞, or the equality should be removed.
  2. [Proof of Lemma 3.4] The sentence 'The union is therefore dense in E∞, and it is contained in the compact set E∞' is not a valid proof of equality; it only establishes density. Please repair this logical gap, for instance by stating the result as a density statement rather than an equality.
  3. [Proposition 3.1 / Theorem 1.3] The proof of Theorem 1.1 depends on Stong's bijection theorem, which is cited from [Sto98] without proof. Since this is a published result, the dependence is acceptable, but the authors should specify the exact theorem in [Sto98] being used, so that readers can verify that the statement here matches Stong's.
  4. [Corollary 4.2] The proof uses nearest-point projection onto the convex set J, which is well-defined and 1-Lipschitz only for closed convex sets; the hypothesis should be amended to say that J is a closed convex set with nonempty interior (or a convex body).
  5. [Section 2, first paragraph] The square Q is later used as a closed set; the notation should be the closed square [−1/2,1/2]^2 rather than an open interval.
  6. [Sections 2–3] The construction in Proposition 2.1 is carried out after translating so that Q=[0,1]^2, while Lemma 3.3 and Lemma 3.4 concern the centered E∞. The authors should state explicitly that the Hölder parameterization g is mapped back to the centered model, or that all results are translation-invariant, to avoid confusion.
  7. [General typography] Please correct typographical issues, such as the spacing in 'Hölder' and the use of 'α Hölder', and ensure that all displayed equations are properly formatted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is self-contained apart from Stong's external bijection theorem, which is not equivalent to the continuous result.

full rationale

The paper's central claim is a continuous surjection from [0,1]^m onto [0,1]^{m+1} with sharp Holder exponent m/(m+1). The derivation chain is self-contained except for Proposition 3.1, which cites Stong's lattice bijection F: X_m^m -> Z^{m+1} from [Sto98]. This is an external theorem by a different author, not a self-citation, and it is not equivalent to the continuous target result. Stong's theorem supplies only a bijection between integer lattice points; the paper independently proves an explicit alpha-Holder parameterization g of the self-similar set E_infty (Proposition 2.1), proves bi-Lipschitz bounds for the linear map L (Proposition 3.2), and assembles the final map as f = T composed with pi composed with L composed with (g,...,g). The surjectivity argument in Lemma 3.5 uses only the bijectivity of F and the lower bound ||x||_infty <= M||F(x)||_infty; it does not assume the existence of the continuous map it is proving. The Holder constant in Theorem 1.1 is derived from explicit estimates in Lemmas 2.4, 2.5, Proposition 2.1, and Proposition 3.2, rather than fitted to the desired conclusion. The volume upper bound and Shchepin's near-critical maps are external benchmarks; the attainability claim is not fed into either. No fitted parameters are renamed as predictions, no load-bearing self-citation chain appears, and no equation is defined in terms of the result it is supposed to derive. Dependence on Stong's unproved-in-this-paper bijection is a correctness/external-support concern, not circularity, because the cited result is prior independent work and the continuous extension remains a nontrivial achievement.

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

All objects in the construction, such as the X-shaped fractal E∞ and the auxiliary triangle set V∞, are explicitly defined rather than postulated. The only externally imported nontrivial result is Stong's discrete bijection theorem. There are no fitted constants or empirical inputs.

assumptions (4)
  • standard math Stong's theorem: for all integers 1≤m≤n, there is an (m/n)-Hölder bijection from Z^m onto Z^n; in particular F: X_m^m → Z^{m+1} is a bijection.
    Used as a black box in Proposition 3.1 and in Lemma 3.5. Cited to [Sto98], not proved in the paper.
  • standard math Hutchinson's theorem for self-similar iterated function systems with the open set condition.
    Justifies existence and uniqueness of the attractors E∞ and V∞, and the identification W=E∞ in Proposition 2.1.
  • standard math Hata's theorem on connectedness of certain self-similar sets.
    Invoked to assert that E∞ is connected; this is background and not central to the main estimate.
  • standard math McShane extension theorem for Lipschitz functions on metric spaces.
    Used in Corollary 4.1 to lift Hölder maps from separated cubes to all of R^m.

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Pith. "Pith review of Space-filling surfaces: sharp H\"older continuous parameterizations from squares to cubes." pith.science (2026). https://pith.science/paper/OYYV3B7J

@misc{pith2026260821246,
  author       = {Pith},
  title        = {Pith review of: Space-filling surfaces: sharp H\"older continuous parameterizations from squares to cubes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OYYV3B7J}},
  note         = {Machine review of arXiv:2608.21246}
}
abstract

Following a hint of Semmes, we employ Stong's bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each $m\geq 2$ we build $\alpha$-H\"older continuous parameterizations $f:[0,1]^m\rightarrow[0,1]^{m+1}$ with sharp exponent $\alpha=m/(m+1)$. In particular, there exist $(2/3)$-H\"older continuous surjections from squares to cubes. This solves Arnold's problem 1988--5.

Figures

Figures reproduced from arXiv: 2608.21246 by the authors.

Figure 2.1
Figure 2.1. From left to right, we illustrate the first three iterates E1, E2, and E3 of Q = [− 1 2 , 1 2 ] 2 under the IFS E when m = 2. side length 1/M, indexed by (p, q) ∈ {0, . . . , M − 1} 2 . We keep the 2M subsquares that intersect one of the two diagonals of Q and throw out the rest, i.e., we keep Qp,q provided that (p, q) belongs to (2.1) A := {(p, q) ∈ {0, . . . , M − 1} 2 : p = q or p + q = M − 1}. Let ϕp,q denote th… view at source ↗
Figure 2.2
Figure 2.2. First two levels of the triangle system when m = 2. The unions of the hypotenuses are the images of s1 and s2. Let q1 := (0, 0) and q2 := (1, 0), and let T := ∆[q1, q2], whose third vertex is ( 1 2 , 1 2 ). Put K := M/2 and p0 := q1, and for 0 ≤ j < M let p2j+1 :=  j+1 M , j M  , p2j+2 :=  j+1 M , j+1 M  (j < K), p2j+1 :=  j M , M−1−j M  , p2j+2 :=  j+1 M , M−1−j M  (j ≥ K). Consecutive points differ by 1/M … view at source ↗
Figure 2.3
Figure 2.3. Graphs of ξ (the first coordinate of s, drawn using its level 3 approximation) and ξb when m = 2. • i = 2j+1 with j ≥ K. Then ξk+1(x) ≤ (j+ξb(u))/M, and the two branches of ξb(x) call for 2ξb(u) ≤ u + 2 and 4ξb(u) ≤ 2M + 1 + u − 2j. The first holds because ξb≤ 1, and the second because 2ξb(u) ≤ u + 1 and j ≤ M − 1. Now let 0 < x ≤ 1. Taking j = 0 in (2.6) gives ξ(x) = ξ(2Mx)/M for x ≤ 1 2M . The ratio ξ(x)/xα is the… view at source ↗
Figures from the paper (2 more)
Figure 2.4
Figure 2.4. Figure 2.4: The images of s1, s2, and s3 together with their three rotated copies when m = 2: s occupies the lower triangle T, Rs the right, R2 s the upper, and R3 s the left. It remains to identify the image of g. The triangles ∆2j+1 and ∆2j+2 lie in Qj,j when j < K and in Qj,M…
Figure 3.1
Figure 3.1. Figure 3.1: The dilation γ2 carries S2 from Stong’s set onto the corners of the family of squares E2. Illustrated with m = 2. sum, and the multiples of M that remain are carried into position i + 1. Writing σi for the number carried, and formally setting σ−1 := 0, we obtain (3.3…

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