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, 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.

现有文献中的几个结果建立了以下强类型的欧几里得密度定理。这些结果表明，适当维度的欧几里得空间中的每组正上巴拿赫密度都包含规定的有限点配置族的所有足够大元素的等距副本。到目前为止，这种类型的所有结果都讨论了定点配置的线性各向同性膨胀。在本文中，我们开始研究由各向异性膨胀产生的点配置族的类比密度定理，即功率类型依赖于单个参数的族，解释为它们的大小。更具体地说，我们证明了 Bourgain 在单纯形顶点上的结果的非各向同性幂型推广，Lyall 和 Magyar 在矩形框顶点上的结果，以及距离树的结果，这是 Lyall 和 Magyar 的距离图论文的一个特例。本文的另一个动机来源是为源自 Cook、Magyar 和 Pramanik 工作的方法的多功能性以及 Durcik 和本作者最近使用的修改提供了额外的证据。最后，本文的另一个目的是挑选出与上述组合问题相关的各向异性多线性奇异积分算子，因为它们本身就很有趣。以及 Durcik 和本作者最近使用的 Pramanik 及其修改。最后，本文的另一个目的是挑选出与上述组合问题相关的各向异性多线性奇异积分算子，因为它们本身就很有趣。以及 Durcik 和本作者最近使用的 Pramanik 及其修改。最后，本文的另一个目的是挑选出与上述组合问题相关的各向异性多线性奇异积分算子，因为它们本身就很有趣。