We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

中文翻译：

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欧几里得空间中的盒、扩展盒和正上密度集

我们证明了在足够大的维度上具有正上 Banach 密度的集合包含三个特定高维模式的所有足够大的扩张的全等副本。这些模式是：2n一个固定的顶点n维矩形框，相同的顶点扩展为n完成三项算术级数的点，并且相同的顶点扩展为n完成三分角的分数。我们的结果提供了文献中几个欧几里得密度定理的通用概括。