论文标题
分区等级和分区晶格
Partition Rank and Partition Lattices
论文作者
论文摘要
我们引入了一种应用分区级别方法的通用方法，这是TAO的切片等级方法的扩展，以扩展到不是对角线的张量。这是通过将Naslund的独特性指标推广到我们所谓的分区指标来实现的。分区指标的优点是两个方面：它们将指定变量集相等时恒定的张量化量化，即使在更一般的设置中，它们通常可以实质上降低分区等级，而与应用独特的指标相比。我们发现的关键是将分区排名方法与Möbius倒置整合到有限集的分区晶格上。通过此，我们统一了文献中分区等级方法的不同应用。然后，我们使用我们的理论来解决一个有限的领域类似物，从而概括了Hart和Iosevich以及独立Shparlinski的结果。此外，我们概括了Pach等人的工作。在限制尺寸的方面，避免了正确的三角形到避免正确的$ k $ configurations的界限尺寸。
We introduce a universal approach for applying the partition rank method, an extension of Tao's slice rank polynomial method, to tensors that are not diagonal. This is accomplished by generalizing Naslund's distinctness indicator to what we call a partition indicator. The advantages of partition indicators are two-fold: they diagonalize tensors that are constant when specified sets of variables are equal, and even in more general settings they can often substantially reduce the partition rank as compared to when a distinctness indicator is applied. The key to our discoveries is integrating the partition rank method with Möbius inversion on the lattice of partitions of a finite set. Through this we unify disparate applications of the partition rank method in the literature. We then use our theory to address a finite field analogue of a question of Erdős, thereby generalizing results of Hart and Iosevich and independently Shparlinski. Furthermore we generalize work of Pach, et al. on bounding sizes of sets avoiding right triangles to bounding sizes of sets avoiding right $k$-configurations.
