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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

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