Large grid subsets without many cospherical points

Abstract

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 $n/4$ toward the Erd\H{o}s--Purdy problem due to Thiele in 1995. For $d \ge 3$, this improves the recent $\Omega \bigl( n^{\frac{3}{d+1}-o(1)} \bigr)$ bound due to Suk and White, confirming their conjectured $\Omega \bigl( n^{\frac{d}{d+1}} \bigr)$ bound in a strong sense, and asymptotically resolves the generalized Erd\H{o}s--Purdy problem posed by Brass, Moser, and Pach.