A note on the no-$(d+2)$-on-a-sphere problem

Andrew Suk; Ethan Patrick White

arxiv: 2412.02866 · v1 · pith:QWYMVOUGnew · submitted 2024-12-03 · 🧮 math.CO · cs.DM

A note on the no-(d+2)-on-a-sphere problem

Andrew Suk , Ethan Patrick White This is my paper

classification 🧮 math.CO cs.DM

keywords fracbestboundconstructcubedimensionalfixedhyperplane

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

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