Rational normal curves as no-$(d+2)$-on-$Q$-quadric sets
Dávid R. Szabó
TL;DR
The paper advances the sharp no-$(d+2)$-on-sphere/quadric problem by constructing large subsets of $[n]^d$ with controlled intersections against affine hyperplanes and $Q$-quadrics, achieving size $n-o(n)$ for all $d\ge 2$ and any nonzero quadratic form $Q$. Central to the approach is a deliberately chosen rational normal curve (via a Veronese embedding and a projective transform) whose affine part lies outside all $Q$-quadrics, enabling Bézout-based bounds that cap affine-hyperplane intersections at $d$ and quadric intersections at $d+1$. The authors also develop finite-field analogues, showing that $Q$-generic sets of size $|\,\mathbb{F}\,|+1-d$ exist under suitable conditions, and then lift these constructions to finite grids in $\mathbb{R}^d$. Overall, the work unifies and extends prior strategies (Thiele, DongXu) under a single geometric framework, yielding near-optimal lower bounds and broad applicability to quadrics beyond spheres with potential implications for incidence geometry and finite-field combinatorics.
Abstract
For every $d\geq 2$, we construct a subset $D\subseteq \{1,2,\dots,n\}^d$ of size $n-o(n)$ such that every affine hyperplane of $\mathbb{R}^d$ intersects $D$ in at most $d$ points, and every hypersphere of $\mathbb{R}^n$ intersects $D$ in at most $d+1$ points. This construction is the largest one currently known, and strongly builds on ideas of Dong, Xu, and also of Thiele. More generally, we prove that the role of hyperspheres can be replaced by $Q$-quadrics, i.e. by quadratic surfaces given by an equation whose degree two homogeneous part equals a fixed quadratic form $Q$. We formulate analogous statements in affine spaces over (finite) fields. Essentially, every construction is given by a suitable rational normal curve in a $d$-dimensional projective space.