On k ‐point configuration sets with nonempty interior
On k ‐point configuration sets with nonempty interior
复制标题
内部非空的 k 点配置集
DOI：
10.1112/mtk.12114
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发表时间：
2022
期刊：
影响因子：
0.8
通讯作者：
Taylor, Krystal
中科院分区：
文献类型：
--
作者：
Greenleaf, Allan;Iosevich, Alex;Taylor, Krystal
我们给出了薄集的叉点配置集具有非空内部的条件，适用于各种不同的配置。这是我们先前工作的延续(J.Geom)。Anal.31(2021)，6662-6680)，推广了Mattila和Sjölin(Math.Nachr.204(1999)，157-162)。我们证明了对于一类一般的K点配置，AK元组集合的配置集e1，⋯，Ek$E_1，\dots，\，E_k$具有非空的内部，只要它们的Hausdorff维度之和满足一个下界，这个下界是由优化相关的广义Radon变换的L2-Soblev估计所决定的。我们用大量的具体例子来说明一般定理。三点构型的应用包括R2$\mathbb{R}^2$中三角形的面积或其外接圆的半径；R3$\mathbb{R}^3$中钉扎的平行四面体的体积；R2$\mathbb{R}^2$和R3$\mathbb{R}^3$中的钉扎距离之比。四点构型的结果包括R$\mathbb{R}$上的交叉比，R2$\mathbb{R}^2$中由四边形确定的三角形面积对，以及RD$\mathbb{R}^d$中差值的点积。
We give conditions fork‐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 (J. Geom. Anal.31(2021), 6662–6680) on 2‐point configurations, extending a theorem of Mattila and Sjölin (Math. Nachr.204(1999), 157–162) for distance sets in Euclidean spaces. We show that for a general class ofk‐point configurations, the configuration set of ak‐tuple of sets, E1,⋯,Ek$E_1,\,\dots ,\, E_k$, has nonempty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizingL2‐Sobolev estimates of associated generalized Radon transforms over all nontrivial partitions of thekpoints into two subsets. We illustrate the general theorems with numerous specific examples. Applications to 3‐point configurations include areas of triangles in R2$\mathbb {R}^2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in R3$\mathbb {R}^3$; and ratios of pinned distances in R2$\mathbb {R}^2$ and R3$\mathbb {R}^3$. Results for 4‐point configurations include cross‐ratios on R$\mathbb {R}$, pairs of areas of triangles determined by quadrilaterals in R2$\mathbb {R}^2$, and dot products of differences in Rd$\mathbb {R}^d$.