On sets defining few ordinary solids

TL;DR

This paper investigates the structure of point sets in four-dimensional space with few ordinary solids, showing that most points lie on a specific algebraic surface, and explores the existence of special configurations with fewer such solids.

Contribution

It establishes a structural characterization of point sets with few ordinary solids in four dimensions and links these configurations to algebraic curves and group structures.

Findings

Most points lie on the intersection of five quadrics.

Sets with fewer ordinary solids relate to finite subgroups of elliptic curves.

Provides bounds on the number of solids incident to exactly four points.

Abstract

Let $\mathcal{S}$ be a set of $n$ points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of $\mathcal{S}$ is less than $Kn^3$ for some $K=o(n^{\frac{1}{7}})$ then, for $n$ sufficiently large, all but at most $O(K)$ points of $\mathcal{S}$ are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size $n$ of an elliptic curve which span less than $\frac{1}{6}n^3$ solids containing exactly four points of $\mathcal{S}$.