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

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 improves the previously best known bound of $\mathrm{\Omega }(n^{\frac{1}{d-1}})$ due to Thiele from 1995.