Non-Euclidean Erdős-Anning Theorems

[2401.06328] Non-Euclidean Erdős-Anning Theorems Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Metric Geometry arXiv:2401.06328 (math) [Submitted on 12 Jan 2024 ( v1 ), last revised 24 Feb 2025 (this version, v3)] Title: Non-Euclidean Erdős-Anning Theorems Authors: David Eppstein View a PDF of the paper titled Non-Euclidean Erd\H{o}s-Anning Theorems, by David Eppstein View PDF HTML (experimental) 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~$\delta$, 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. Comments: 27 pages, 13 figures. This is the full version of a paper to appear in shortened form in the proceedings of the 41st International Symposium on Computational Geometry (SoCG 2025) Subjects: Metric Geometry (math.MG) ; Computational Geometry (cs.CG) Cite as: arXiv:2401.06328 [math.MG] (or arXiv:2401.06328v3 [math.MG] for this version) https://doi.org/10.48550/arXiv.2401.06328 Focus to learn more arXiv-issued DOI via DataCite Journal reference: J. Computational Geometry 17 (2): 46-76, 2026 Related DOI : https://doi.org/10.20382/jocg.v17i2a2 Focus to learn more DOI(s) linking to related resources Submission history From: David Eppstein [ view email ] [v1] Fri, 12 Jan 2024 02:41:06 UTC (2,422 KB) [v2] Mon, 22 Jan 2024 01:28:05 UTC (2,423 KB) [v3] Mon, 24 Feb 2025 01:54:00 UTC (2,425 KB) Full-text links: Access Paper: View a PDF of the paper titled Non-Euclidean Erd\H{o}s-Anning Theorems, by David Eppstein View PDF HTML (experimental) TeX Source view license Current browse context: math.MG < prev | next > new | recent | 2024-01 Change to browse by: cs cs.CG math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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