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DOI: 10.4230/LIPIcs.SoCG.2025.76

A Note on the No-(d+2)-On-a-Sphere Problem

Authors: Andrew Suk and Ethan Patrick White

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

For fixed d ≥ 3, we construct subsets of the d-dimensional lattice cube [n]^d of size n^{3/(d + 1) - o(1)} with no d+2 points on a sphere or a hyperplane. This improves the previously best known bound of Ω(n^{1/(d-1)}) due to Thiele from 1995.

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Andrew Suk and Ethan Patrick White. A Note on the No-(d+2)-On-a-Sphere Problem. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 76:1-76:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{suk_et_al:LIPIcs.SoCG.2025.76,
  author =	{Suk, Andrew and White, Ethan Patrick},
  title =	{{A Note on the No-(d+2)-On-a-Sphere Problem}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{76:1--76:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.76},
  URN =		{urn:nbn:de:0030-drops-232287},
  doi =		{10.4230/LIPIcs.SoCG.2025.76},
  annote =	{Keywords: General position, no-four-on-a-cirle, d-dimensional lattice cube}
}

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A Note on the No-(d+2)-On-a-Sphere Problem

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