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

Authors Andrew Suk, Ethan Patrick White

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ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • General position
  • no-four-on-a-cirle
  • d-dimensional lattice cube

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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) https://doi.org/10.4230/LIPIcs.SoCG.2025.76

Author Details

Andrew Suk
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA
Ethan Patrick White
  • Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL, USA

Funding

  • Suk, Andrew: Supported by NSF CAREER award DMS-1800746, NSF award DMS-1952786, and NSF grant DMS-2246847.
  • White, Ethan Patrick: supported by an NSERC PDF Award.

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