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.

翻译：对于固定的$d\geq 3$，我们在$d$维格点立方体$[n]^d$中构造了规模为$n^{\frac{3}{d + 1} - o(1)}$的子集，该子集不存在$d+2$个点共球或共超平面。这一结果改进了Thiele于1995年提出的已知最佳下界$\Omega(n^{\frac{1}{d-1}})$。