论文标题
在$ k $ - 点配置集，具有非空内部
On $k$-point configuration sets with nonempty interior
论文作者
论文摘要
我们提供$ k $ - 点配置的薄套件的条件，以具有非空的内饰，适用于各种配置。这是我们较早的工作\ cite {git19}在两点配置上的延续，扩展了Mattila和Sjölin\ cite {MS99}的定理，以用于欧几里得空间中的距离集。 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 $点分为两个子集。我们用许多具体示例说明了一般定理。 3点配置的应用程序包括$ \ Mathbb r^2 $中的三角形区域或其限制圈子的半径； $ \ mathbb r^3 $中的固定并行牵引的卷； $ \ Mathbb r^2 $和$ \ Mathbb r^3 $中固定距离的比率。 4点配置的结果包括$ \ mathbb r $上的交叉比例，由$ \ Mathbb r^2 $确定的三角形区域对，以及$ \ mathbb r^d $中的差异产品。
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 $\mathbb R^d$.
