具有非空内部的$k$ -点配置集
On $k$-point configuration sets with nonempty interior
摘要：
我们给出了薄集的$k$ -点配置集具有非空内部的条件，适用于各种配置。这是我们早期工作的延续\cite{GIT19}上的2点配置，扩展了一个定理的马蒂拉和Sjölin \cite{MS99}的距离集在欧氏空间。我们发现，对于一般类的$k$ -点配置，配置集的$k$ -元组的集合，$E_1,\,\dots,\, E_k$，具有非空的内部提供的Hausdorff维数的总和满足一个下限，规定通过优化$L^2$ -Sobolev估计相关的广义拉东变换在所有非平凡分区的$k$点成两个子集。我们用许多具体的例子来说明一般定理。三点构型的应用包括$\mathbb R^2$中三角形的面积或其外接圆的半径; $\mathbb R^3$中钉扎平行六面体的体积;以及$\mathbb R^2$和$\mathbb R^3$中钉扎距离的比值。4点配置的结果包括$\mathbb R$上的交比、由$\mathbb R^2$中的四边形确定的三角形面积对以及$\mathbb R^d$中差异的点积。
英文摘要：
We give conditions for $k$-point configuration sets of thin sets to have nonempty interior, applicable to a wide variety of configurations. This is a continuation of our earlier work \cite{GIT19} on 2-point configurations, extending a theorem of Mattila and Sjölin \cite{MS99} for distance sets in Euclidean spaces. We show that for a general class of $k$-point configurations, the configuration set of a $k$-tuple of sets, $E_1,\,\dots,\, E_k$, has nonempty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizing $L^2$-Sobolev estimates of associated generalized Radon transforms over all nontrivial partitions of the $k$ points into two subsets. We illustrate the general theorems with numerous specific examples. Applications to 3-point configurations include areas of triangles in $\mathbb R^2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in $\mathbb R^3$; and ratios of pinned distances in $\mathbb R^2$ and $\mathbb R^3$. Results for 4-point configurations include cross-ratios on $\mathbb R$, triangle area pairs determined by quadrilaterals in $\mathbb R^2$, and dot products of differences in $\mathbb R^d$.
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