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

TL;DR

This paper constructs large subsets within a d-dimensional lattice cube that avoid having d+2 points on a sphere or hyperplane, improving previous bounds and advancing understanding of geometric configurations in discrete spaces.

Contribution

The authors provide a new construction of large point sets in high-dimensional cubes avoiding certain geometric configurations, surpassing earlier known bounds.

Findings

Constructed subsets of size approximately n^{3/(d+1)} with no d+2 points on a sphere or hyperplane

Improved the lower bound from (n^{1/(d-1)}) to n^{3/(d+1)}

Advances the understanding of extremal configurations in discrete geometry

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.