Let $F$ be a Bedford-McMullen carpet defined by independent exponents. We prove that $\overline{\dim}_B (\ell \cap F) \leq \max \lbrace \dim^* F -1,0 \rbrace$ for all lines $\ell$ not parallel to the principal axes, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford-McMullen carpets, that is, carpets $F$ and $E$ such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self affine measures on them are mutually singular, and prove that they do not embed affinely into each other. We obtain these results as an application of a slicing Theorem for products of certain Cantor sets. This Theorem is a generlization of the results of Shmerkin and Wu, that proved Furstenberg's slicing Conjecture.

中文翻译：

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自仿地毯的切片定理和刚度现象

假设$ F $是由独立指数定义的Bedford-McMullen地毯。我们证明$ \ overline {\ dim} _B（\ ell \ cap F）\ leq \ max \ lbrace \ dim ^ * F -1,0 \ rbrace $对于所有不平行于主轴的线$ \ ell $，其中$ \ dim ^ * $是Furstenberg的星形尺寸（微装置的最大尺寸）。我们还证明了贝德福德-麦克穆伦地毯无法比拟的几个刚性结果，即地毯$ F $和$ E $使得所有定义的指数都是独立的：假设各种条件，我们找到了此类地毯相交的尺寸范围，请参见它们上的自我仿射度量是互为奇异的，并证明它们没有彼此仿射嵌入。我们将这些结果作为切片定理对某些Cantor集产品的应用。这个定理是Shmerkin和Wu的结果的泛化，