论文标题
通过飞机上的点配置跨越的区域
Areas spanned by point configurations in the plane
论文作者
论文摘要
我们使用\ cite {groupAction}中引入的组操作框架在$（k+1）$ - 点配置上考虑了$（k+1）$ - 点配置的过度确定的猎鹰类型问题。我们将平面中的$（k+1）$ - 点配置定义为$ \ r^{\ binom {\ binom {k+1} {2}} $，并在配置中由每对点跨越的平行四边形区域给出的条目。我们表明，所有区域类型的空间均为$ 2K-1 $尺寸，并证明了一个紧凑的集合$ e \ subset \ r^d $的足够大的Hausdorff尺寸决定了一个面积类型的效率。
We consider an over-determined Falconer type problem on $(k+1)$-point configurations in the plane using the group action framework introduced in \cite{GroupAction}. We define the area type of a $(k+1)$-point configuration in the plane to be the vector in $\R^{\binom{k+1}{2}}$ with entries given by the areas of parallelograms spanned by each pair of points in the configuration. We show that the space of all area types is $2k-1$ dimensional, and prove that a compact set $E\subset\R^d$ of sufficiently large Hausdorff dimension determines a positve measure set of area types.
