论文标题
通过微局部分区优化的配置集的非空内部
Nonempty interior of configuration sets via microlocal partition optimization
论文作者
论文摘要
我们证明了Mattila-Sjölin类型的新结果，在薄设置的Hausdorff尺寸上给出了$ e \ subset \ bbb r^d $的下限，以确保由$ e $的元素生成的各种$ k $ - 点配置集，具有非空的内部装置。我们以前的工作中的尺寸阈值\ cite {git20}是通过与配置函数相关联的一家广义ra换变换的家族来决定的，然后优化了$ l^2 $ -sobolev估计它们的估计，而不是$ k $点的所有非平凡的双党分区。在当前工作中，我们通过允许优化在配置的起点关系上，甚至在入射关系关系的孔子捆绑上进行优化来扩展这一点。我们使用这种方法来证明（i）在集合中由四边形和五角星确定的子三角形区域的Mattila-Sjölin类型结果。 （ii）$ \ bbb r^d $中4个距离的距离的比率对； （iii）$ \ bbb r^d $中三角形的相似性类别，以及（iv）提供了palsson和Romero Acosta在$ \ bbb r^d $中的一致性三角形类别的简短证明。
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 $\Bbb R^d$.
