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

Andrew Suk, Ethan Patrick White

For fixed $d\ge 3$, 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 im…