各向异性点构形的密度定理
Density theorems for anisotropic point configurations
摘要：
现有文献中的几个结果建立了下列强类型的欧几里得密度定理。这些结果表明，在适当维数的欧几里得空间中，每一个正的上Banach密度集都包含一个给定的有限点构形族的所有足够大的元素的等距拷贝.到目前为止，这类结果都是讨论不动点位形的线性各向同性膨胀体。在本文中，我们开始研究由各向异性膨胀产生的点构形族的类似密度定理，即，具有幂型依赖于单个参数的族，该参数被解释为它们的大小。更具体地说，在这里，我们证明了非各向同性的权力型推广的结果Bourgain顶点的单纯形，一个结果Lyall和Magyar顶点的矩形框，和一个结果的距离树，这是一个特殊情况下的论文距离图Lyall和Magyar。本文的另一个动机来源是为库克、马扎尔和普拉马尼克的工作以及杜西克和本文作者最近使用的修改方法的多功能性提供了额外的证据。最后，本文的另一个目的是挑出与上述组合问题相关的各向异性多线性奇异积分算子，因为它们本身是有趣的。
英文摘要：
Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, here we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.
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