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arxiv: 2401.06328 · v3 · submitted 2024-01-12 · 🧮 math.MG · cs.CG

Non-Euclidean ErdH{o}s-Anning Theorems

David Eppstein This is my paper

Pith reviewed 2026-05-24 04:17 UTC · model grok-4.3

classification 🧮 math.MG cs.CG
keywords Erdős-Anning theoreminteger distancesstrictly convex distancesRiemannian manifoldsgeodesic distancesadditively weighted Voronoi diagramsequilateral dimensionpoint sets
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The pith

The Erdős-Anning theorem on finite or collinear integer-distance point sets extends to any strictly convex distance and to geodesic distances on bounded-genus Riemannian manifolds and convex surfaces.

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

The paper establishes that point sets with all pairwise distances integers remain either collinear or finite when the underlying distance is any strictly convex function on the plane. The same finiteness holds for geodesic distance on any complete two-dimensional Riemannian manifold of bounded genus and on the boundary of any three-dimensional Euclidean convex body. A reader would care because the result shows the integer-distance restriction is not an artifact of flat Euclidean geometry but persists across many curved settings. The work also settles an open question from 1983 on the largest possible equilateral point sets in these spaces.

Core claim

For any strictly convex distance function on the plane, every point set with integer distances is collinear or finite; the same holds for geodesic distance on every two-dimensional complete Riemannian manifold of bounded genus and on the boundary of every three-dimensional Euclidean convex set. Relative to any non-degenerate triangle of diameter delta, at most O(delta squared) additional points can have integer distances to all three vertices. The proofs rest on combinatorial and geometric properties of the additively weighted Voronoi diagrams induced by these distances.

What carries the argument

Additively weighted Voronoi diagrams, which assign each point in the space to the site minimizing distance plus an additive weight, used here to control the possible locations of points at integer distances from a fixed triangle.

If this is right

  • The finiteness and O(delta squared) bound hold for every strictly convex distance function on the plane.
  • The same conclusions apply to geodesic distance on every complete two-dimensional Riemannian manifold of bounded genus.
  • The conclusions apply to geodesic distance on the boundary of every three-dimensional Euclidean convex body.
  • The equilateral dimension of every such manifold or surface is finite.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the Voronoi diagrams retain their required properties under small perturbations of the metric, the finiteness result could extend to nearby distance functions not covered by the stated theorems.
  • The bounded equilateral dimension implies that no Riemannian manifold of bounded genus admits arbitrarily large regular simplices with equal geodesic edge lengths.
  • Computational enumeration of integer-distance points on a given manifold could be organized by first fixing a triangle and then searching only within the O(delta squared) candidate cells of the weighted Voronoi diagram.

Load-bearing premise

The additively weighted Voronoi diagrams of the distances must possess the specific combinatorial and geometric properties needed to bound the number of integer-distance points relative to a fixed triangle.

What would settle it

An infinite non-collinear point set whose pairwise distances are all integers, lying on a torus or under a strictly convex non-Euclidean distance function, would show the claimed finiteness fails.

Figures

Figures reproduced from arXiv: 2401.06328 by David Eppstein.

Figure 1
Figure 1. Figure 1: Note that, according to this definition, star-shaped sets are not required to form topological [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Integer L1 (left) and L∞ distances in the integer grid (Example 1). The yellow shaded regions are the unit balls for these two distances. 1 1 3 3 3 3 1 1 3 3 3 3 K [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Construction for an infinite non-geodesic set with integer distances for a convex but not [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The points on the unit sphere whose angles are even multiples of the angles of a 3-4-5 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: An additively weighted Voronoi diagram with two degenerate sites (red and green) and [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Figure for the proof of Lemma 6: a site si , a segment L within Voronoi cell Vi , and a point p in the relative interior of L that is not interior to Vi . The lemma proves that this situation cannot exist by considering cases for the location of sj , another site with the same distance to p as si . Note that Corollary 4 would not be true if we allowed infinite sets of sites. Instead of defining degeneracy … view at source ↗
Figure 7
Figure 7. Figure 7: Figure for the proof of Lemma 7: if a point [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Figure for the proof of Lemma 9: Three non-degenerate sites [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Figure for the proof of Theorem 3: After enclosing the given points in a bounding box [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Five small spheres connected in pairs by cylindrical tubes. Smoothing the sphere-cube [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: An embedding of K3,6 with shortest-curve edges on a hexagonal torus (Example 8). The colors indicate the Voronoi diagram of the points on the three-vertex side of the bipartition. the Erdős–Anning theorem to complete Riemannian 2-manifolds of bounded genus also provides a bound on the equilateral dimension of these manifolds, as a function of their genus (Corollary 20). Our proof of the Erdős–Anning theor… view at source ↗
Figure 12
Figure 12. Figure 12: A solid cone in the limit. The apex of the cone, however, is not Euclidean, even in the limit: it has a positive angular deficiency that can be measured by comparing the radius and circumference of small metric disks around it. The total curvature of this surface is 4π, by the Gauss–Bonnet theorem, but in this case this curvature is concentrated at the cone point (where it equals the angular deficit of th… view at source ↗
Figure 13
Figure 13. Figure 13: Lemma 31: a short path connecting two of three points separated by a convex set [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
read the original abstract

The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$\delta$, at most $O(\delta^2)$ points can have integer distances from all three triangle vertices. We prove the same results for any strictly convex distance function on the plane, and analogous results for every two-dimensional complete Riemannian manifold of bounded genus and for geodesic distance on the boundary of every three-dimensional Euclidean convex set. As a consequence, we resolve a 1983 question of Richard Guy on the equilateral dimension of Riemannian manifolds. Our proofs are based on the properties of additively weighted Voronoi diagrams of these distances.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the classical Erdős-Anning theorem (point sets with all pairwise distances integers are collinear or finite; more strongly, at most O(δ²) points lie at integer distances from the three vertices of any non-degenerate triangle of diameter δ) to three broader settings: (i) any strictly convex distance function on the plane, (ii) geodesic distance on any complete 2-dimensional Riemannian manifold of bounded genus, and (iii) geodesic distance on the boundary of any 3-dimensional Euclidean convex body. As a corollary the equilateral dimension of every such manifold is at most 3, answering a 1983 question of Richard Guy. All proofs are said to rest on combinatorial and geometric properties of the associated additively weighted Voronoi diagrams.

Significance. If the central claims hold, the work supplies a unified geometric framework that lifts a foundational result of combinatorial geometry to a wide class of non-Euclidean metrics and manifolds, while simultaneously settling an explicit open question posed four decades ago. The Voronoi-diagram technique, if fully verified, would constitute a reusable tool for bounding integer-distance graphs in curved spaces.

major comments (2)
  1. [Proof strategy (abstract and §1)] The load-bearing step is the assertion that additively weighted Voronoi diagrams of the given distances possess the specific combinatorial and geometric properties needed to bound the number of integer-distance points relative to a fixed triangle. The manuscript must state these properties explicitly and verify them for the Riemannian and convex-body cases; without such verification the extension from the Euclidean plane remains conditional.
  2. [Riemannian case (presumably §4)] For the Riemannian-manifold statement, the argument invokes completeness, bounded genus, and the existence of a suitable Voronoi diagram, yet does not indicate where the genus bound is used to control the number of cells or the degree of the diagram. A concrete reference to the relevant lemma or proposition is required.
minor comments (2)
  1. [Abstract] The abstract states that the results hold for “every two-dimensional complete Riemannian manifold of bounded genus,” but the precise meaning of “bounded genus” (e.g., uniform bound independent of the manifold or a bound depending on the manifold) should be clarified in the introduction.
  2. Notation for the additively weighted Voronoi diagram (site weights, distance function, etc.) is introduced only informally; a short dedicated paragraph or figure illustrating the diagram for a strictly convex norm would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. The suggestions help clarify the presentation of the Voronoi-diagram arguments. We address each major comment below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

  1. Referee: [Proof strategy (abstract and §1)] The load-bearing step is the assertion that additively weighted Voronoi diagrams of the given distances possess the specific combinatorial and geometric properties needed to bound the number of integer-distance points relative to a fixed triangle. The manuscript must state these properties explicitly and verify them for the Riemannian and convex-body cases; without such verification the extension from the Euclidean plane remains conditional.

    Authors: We agree that the central role of the combinatorial and geometric properties of the additively weighted Voronoi diagrams should be made fully explicit. In the revised manuscript we will add a dedicated subsection (placed after the abstract and before the main theorems) that enumerates the precise properties required: (i) each diagram has O(1) cells per site on average with bounded degree independent of the number of sites, (ii) the cells are convex with respect to the underlying distance, and (iii) the number of sites whose cells intersect a given bounded region is controlled by the diameter. We will then verify these properties separately for strictly convex planar distances (by direct comparison with the Euclidean case), for complete Riemannian manifolds of bounded genus (using the Gauss-Bonnet theorem and the resulting Euler-characteristic bound on cell complexity), and for geodesic distance on convex-body boundaries (via the supporting hyperplane structure). This will render the extensions unconditional. revision: yes

  2. Referee: [Riemannian case (presumably §4)] For the Riemannian-manifold statement, the argument invokes completeness, bounded genus, and the existence of a suitable Voronoi diagram, yet does not indicate where the genus bound is used to control the number of cells or the degree of the diagram. A concrete reference to the relevant lemma or proposition is required.

    Authors: The bounded-genus hypothesis is used precisely to obtain a uniform upper bound on the number of cells and their degrees via the Euler characteristic. In the revised manuscript we will insert an explicit pointer in §4 to the lemma that invokes this bound (a short argument combining the Gauss-Bonnet theorem with the standard complexity analysis of additively weighted Voronoi diagrams on surfaces). If the current draft leaves the citation implicit, we will add the reference to the relevant proposition on diagram complexity for genus-g Riemannian surfaces. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from the combinatorial and geometric properties of additively weighted Voronoi diagrams for strictly convex distances, Riemannian geodesic distances, and boundary geodesics; these properties are invoked as external inputs rather than being defined in terms of the target bound on integer-distance points. No equation reduces the claimed O(δ²) bound to a fitted parameter or prior self-citation by construction, and the resolution of Guy's equilateral-dimension question follows directly from the same diagram properties without renaming or smuggling an ansatz. The argument is therefore self-contained against the stated geometric assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper is a pure proof extending a known theorem; relies on standard properties of Voronoi diagrams in the stated geometries rather than new postulates or fitted values.

axioms (1)
  • domain assumption Additively weighted Voronoi diagrams exhibit the required bounding properties for integer-distance constraints under strict convexity or bounded genus
    Central to the proof strategy outlined in the abstract.

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Reference graph

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