Large grid subsets without many cospherical points

Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n{]}^d$ of size $n-o(n)$ with no $d+2$ points on a sphere or a hyperplane. For $d=2$, this improves the previously best known lower bound of $n\mathrm{/}4$ toward the Erdős–Purdy problem due to Thiele in 1995. For $d\ge 3$, this improves the recent $\mathrm{\Omega }\left(n^{\frac{3}{d+1}-o(1)}\right)$ bound due to Suk and White, confirming their conjectured $\mathrm{\Omega }\left(n^{\frac{d}{d+1}}\right)$ bound in a strong sense, and asymptotically resolves the generalized Erdős–Purdy problem posed by Brass, Moser, and Pach.