Improved versions of some Furstenberg type slicing Theorems for self-affine carpets

Abstract

Let F be a Bedford-McMullen carpet defined by independent integer exponents. We prove that for every line ℓ ⊆ R 2 not parallel to the major axes,andwhere dim * is Furstenberg's star dimension (maximal dimension of microsets). This improves the state of art results on Furstenberg type slicing Theorems for affine invariant carpets.

Slices and distances: on two problems of Furstenberg and Falconer

“…Nevertheless, by modifying the method of Wu, A. Algom [2] proved an upper bound for the dimension of linear slices of McMullen carpets, that reduces to Conjecture 1.1 in the product case. The bound was recently improved further by A. Algom and M. Wu [3], but the optimal result remains elusive.…”