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

Andrew Suk; Ethan Patrick White

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

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arxiv 2412.02866 v1 pith:QWYMVOUG submitted 2024-12-03 math.CO cs.DM

Andrew Suk , Ethan Patrick White This is my paper · ORCID

classification math.COcs.DM

keywords fracbestboundconstructcubedimensionalfixedhyperplane

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abstract

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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On subsets of lattice cubes avoiding affine and spherical degeneracies
math.CO 2025-09 conditional novelty 6.0 of 10

New lower bounds for lattice sets avoiding subspheres and subspaces, including f_circ(n) ≥ 7n/12, via deletion-method counting of cyclic quadrilaterals and cospherical tuples.

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