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arXiv · 2401.06328

Non-Euclidean Erd\H{o}s-Anning Theorems

David Eppstein
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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.

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David Eppstein. 2024-01-12. Non-Euclidean Erd\H{o}s-Anning Theorems. https://doi.org/10.20382/jocg.v17i2a2

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