On sets defining few ordinary solids

Simeon Ball; Enrique Jimenez

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arXiv · 1808.06388

On sets defining few ordinary solids

Simeon Ball·Enrique Jimenez

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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 $\mathcal{S}$.

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Simeon Ball, Enrique Jimenez. 2018-08-20. On sets defining few ordinary solids. https://arxiv.org/abs/1808.06388

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