Slicing theorems and rigidity phenomena for self-affine carpets

Let (Formula presented.) be a Bedford–McMullen carpet defined by independent exponents. We prove that (Formula presented.) for all lines (Formula presented.) not parallel to the principal axes, where (Formula presented.) is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford–McMullen carpets, that is, carpets (Formula presented.) and (Formula presented.) 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 generalization of the results of Shmerkin [Ann. of Math. (Formula presented.) 189 (2019) 319–391] and Wu [Ann. of Math. (Formula presented.) 189 (2019) 707–751], which proved Furstenberg's slicing conjecture [Problems in analysis (ed. R. C. Gunning; Princeton University Press, Princeton, NJ, 1970) 41–59].