通过微局部分割优化证明构型集的非空内部
Nonempty interior of configuration sets via microlocal partition optimization
AI总结：
本文通过在构型关联关系及其余法丛上进行微局部分割优化，改进了 $L^2$-Sobolev 估计，证明了多种 $k$ 点构型集具有非空内部的 Mattila-Sjölin 型结果。
摘要：
我们证明了新的 Mattila-Sjölin 型结果，给出了关于 $E\subset \Bbb R^d$ 的 Hausdorff 维数下界，以确保由 $E$ 的元素生成的各种 $k$ 点构型集具有非空内部。我们之前工作 \cite{GIT20} 中的维数阈值是通过将广义 Radon 变换族关联到构型函数，然后在 $k$ 个点的所有非平凡二部划分上优化其 $L^2$-Sobolev 估计来确定的。在当前工作中，我们对此进行了扩展，允许在构型的关联关系上进行局部优化，甚至在其余法丛上进行微局部优化。我们使用这种方法证明了以下情形的 Mattila-Sjölin 型结果：(i) 由集合 $E\subset\Bbb R^2$ 中的四边形和五边形确定的子三角形面积；(ii) $\Bbb R^d$ 中四元组的距离比对；以及 (iii) $\Bbb R^d$ 中三角形的相似类；并 (iv) 简要证明了 Palsson 和 Romero Acosta 关于 $\Bbb R^d$ 中三角形全等类的结果。
英文摘要：
We prove new results of Mattila-Sjölin type, giving lower bounds on Hausdorff dimensions of thin sets $E\subset \Bbb R^d$ ensuring that various $k$-point configuration sets, generated by elements of $E$, have nonempty interior. The dimensional thresholds in our previous work \cite{GIT20} were dictated by associating to a configuration function a family of generalized Radon transforms, and then optimizing $L^2$-Sobolev estimates for them over all nontrivial bipartite partitions of the $k$ points. In the current work, we extend this by allowing the optimization to be done locally over the configuration's incidence relation, or even microlocally over the conormal bundle of the incidence relation. We use this approach to prove Mattila-Sjölin type results for (i) areas of subtriangles determined by quadrilaterals and pentagons in a set $E\subset\Bbb R^2$; (ii) pairs of ratios of distances of 4-tuples in $\Bbb R^d$; and (iii) similarity classes of triangles in $\Bbb R^d$, as well as to (iv) give a short proof of Palsson and Romero Acosta's result on congruence classes of triangles in $\Bbb R^d$.
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