Rational normal curves as no-$(d+2)$-on-$Q$-quadric sets

TL;DR

This paper constructs large point sets in Euclidean and finite spaces that intersect low-dimensional geometric objects in a limited number of points, using rational normal curves and quadratic forms.

Contribution

It introduces the largest known sets with controlled intersections with hyperplanes, hyperspheres, and quadrics, extending previous ideas with new geometric constructions.

Findings

Constructed large point sets with limited intersections in Euclidean space.

Extended intersection bounds from hyperspheres to general quadrics.

Provided analogous constructions in finite field settings.

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.