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Non-Euclidean Erdős-Anning Theorems

Author David Eppstein

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  • Theory of computation → Computational geometry
Keywords
  • integer distances
  • additively weighted Voronoi diagrams
  • convex distance functions
  • Riemannian manifolds
  • geodesic distance

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Abstract

The Erdős-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 δ, at most O(δ²) 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.

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David Eppstein. Non-Euclidean Erdős-Anning Theorems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.46

Author Details

David Eppstein
  • University of California, Irvine, CA, USA

Funding

  • Eppstein, David: Research supported in part by NSF grant CCF-2212129.

References

  1. Norman H. Anning and Paul Erdős. Integral distances. Bulletin of the American Mathematical Society, 51(8):598-600, 1945. URL: https://doi.org/10.1090/S0002-9904-1945-08407-9.
  2. Peter F. Ash and Ethan D. Bolker. Generalized Dirichlet tessellations. Geometriae Dedicata, 20(2):209-243, 1986. URL: https://doi.org/10.1007/BF00164401.
  3. Jeffrey M. Augenbaum and Charles S. Peskin. On the construction of the Voronoi mesh on a sphere. Journal of Computational Physics, 59(2):177-192, 1985. URL: https://doi.org/10.1016/0021-9991(85)90140-8.
  4. Mikhail Bogdanov, Olivier Devillers, and Monique Teillaud. Hyperbolic Delaunay complexes and Voronoi diagrams made practical. Journal of Computational Geometry, 5(1):56-85, 2014. URL: https://doi.org/10.20382/JOCG.V5I1A4.
  5. L. Paul Chew and Robert L. (Scot) Drysdale, III. Generalization of Voronoi diagrams in the plane. SIAM Journal on Computing, 10(1):73-87, 1981. URL: https://doi.org/10.1137/0210006.
  6. L. Paul Chew and Robert L. (Scot) Drysdale, III. Voronoi diagrams based on convex distance functions. In Joseph O'Rourke, editor, Proceedings of the First Annual Symposium on Computational Geometry, Baltimore, Maryland, USA, June 5-7, 1985, pages 235-244. Association for Computing Machinery, 1985. URL: https://doi.org/10.1145/323233.323264.
  7. Robert Connelly. Convex Polyhedra by A. D. Alexandrov. SIAM Review, 48(1):157-160, March 2006. URL: https://doi.org/10.1137/SIREAD000048000001000149000001.
  8. Olivier Devillers, Stefan Meiser, and Monique Teillaud. The space of spheres, a geometric tool to unify duality results on Voronoi diagrams. In Proceedings of the 4th Canadian Conference on Computational Geometry (CCCG '92), pages 263-268, 1992. URL: https://cccg.ca/proceedings/1992/Paper43.pdf.
  9. Paul Erdős. Integral distances. Bulletin of the American Mathematical Society, 51:996, 1945. URL: https://doi.org/10.1090/S0002-9904-1945-08490-0.
  10. Leonhard Euler. Fragmenta arithmetica ex Adversariis mathematicis deprompta, C: Analysis Diophantea. In Opera postuma, volume I, pages 204-263. Eggers, Petropolis, 1862. Theorem 65, p. 229. URL: https://scholarlycommons.pacific.edu/euler-works/806/.
  11. Rachel Greenfeld, Marina Iliopoulou, and Sarah Peluse. On integer distance sets. Electronic preprint arxiv:2401.10821, 2024. Google Scholar
  12. Richard K. Guy. An olla-podrida of open problems, often oddly posed. The American Mathematical Monthly, 90(3):196-200, 1983. URL: https://doi.org/10.2307/2975549.
  13. Heiko Harborth. Integral distances in point sets. In Charlemagne and his Heritage: 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), pages 213-224. Brepols, Turnhout, Belgium, 1998. URL: https://doi.org/10.1484/M.STHS-EB.4.2017040.
  14. H. Hopf and W. Rinow. Ueber den Begriff der vollständigen differentialgeometrischen Fläche. Commentarii Mathematici Helvetici, 3(1):209-225, 1931. URL: https://doi.org/10.1007/BF01601813.
  15. Christian Icking, Rolf Klein, Ngoc-Minh Lê, and Lihong Ma. Convex distance functions in 3-space are different. Fundamenta Informaticae, 22(4):331-352, 1995. URL: https://doi.org/10.3233/FI-1995-2242.
  16. Menelaos I. Karavelas and Ioannis Z. Emiris. Root comparison techniques applied to computing the additively weighted Voronoi diagram. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA, pages 320-329. ACM and SIAM, 2003. URL: https://dl.acm.org/citation.cfm?id=644108.644161.
  17. Ngoc-Minh Lê. On non-smooth convex distance functions. Information Processing Letters, 63(6):323-329, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00136-1.
  18. Greg Leibon and David Letscher. Delaunay triangulations and Voronoi diagrams for Riemannian manifolds. In Siu-Wing Cheng, Otfried Cheong, Pankaj K. Agarwal, and Steven Fortune, editors, Proceedings of the Sixteenth Annual Symposium on Computational Geometry, Clear Water Bay, Hong Kong, China, June 12-14, 2000, pages 341-349. ACM, 2000. URL: https://doi.org/10.1145/336154.336221.
  19. Jeremy Mann. Equilateral dimension of Riemannian manifolds with bounded curvature. Rose-Hulman Undergraduate Mathematics Journal, 14(1):Article 8, 2013. URL: https://scholar.rose-hulman.edu/rhumj/vol14/iss1/8/.
  20. M. Mazón and T. Recio. Voronoi diagrams on orbifolds. Computational Geometry: Theory & Applications, 8(5):219-230, 1997. URL: https://doi.org/10.1016/S0925-7721(96)00017-X.
  21. R. E. Miles. Random points, sets and tessellations on the surface of a sphere. Sankhyā, Series A, 33:145-174, 1971. URL: https://www.jstor.org/stable/25049720.
  22. S. B. Myers. Arcs and geodesics in metric spaces. Transactions of the American Mathematical Society, 57:217-227, 1945. URL: https://doi.org/10.1090/S0002-9947-1945-0011792-9.
  23. Hyeon-Suk Na, Chung-Nim Lee, and Otfried Cheong. Voronoi diagrams on the sphere. Computational Geometry, Theory and Applications, 23(2):183-194, 2002. URL: https://doi.org/10.1016/S0925-7721(02)00077-9.
  24. Frank Nielsen and Richard Nock. Hyperbolic Voronoi diagrams made easy. In Bernady O. Apduhan, Osvaldo Gervasi, Andrés Iglesias, David Taniar, and Marina L. Gavrilova, editors, Proceedings of the 2010 International Conference on Computational Science and Its Applications, ICCSA 2010, Fukuoka, Japan, March 23-26, 2010, pages 74-80. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/ICCSA.2010.37.
  25. József Solymosi. Note on integral distances. Discrete & Computational Geometry, 30(2):337-342, 2003. URL: https://doi.org/10.1007/s00454-003-0014-7.
  26. Matthias Weber, David Hoffman, and Michael Wolf. An embedded genus-one helicoid. Proceedings of the National Academy of Sciences, 102(46):16566-16568, 2005. URL: https://doi.org/10.1073/pnas.0508008102.
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