关于定义少数普通体的集合
On sets defining few ordinary solids
摘要：
设$\mathcal{S}$是实四维空间中的一组$n$点，它们不共面且跨越整个空间。我们证明了，如果事件与$\mathcal{S}$正好四个点的固体的数量小于$Kn^3$的一些$K=o(n^{\frac{1}{7}})$，然后，$n$足够大，但最多$O(K)$点的$\mathcal{S}$包含在五个线性无关二次曲面的交叉。相反，我们证明，有有限的子群的大小$n$的椭圆曲线的跨度小于$\frac{1}{6}n^3$固体恰好包含四个点的$\mathcal{S}$。
英文摘要：
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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