Large slices through self affine carpets

Abstract

Let $F\subseteq [0,1{]}^2$ be a Bedford-McMullen carpet defined by exponents $m>n$, that projects to $[0,1]$ on the $y$-axis. We show that under mild conditions on $F$, there are many non principle lines $\ell$ such that ${\dim }^{\ast }F\cap \ell ={\dim }^{\ast }F-1$, where ${\dim }^{\ast }$ is Furstenberg's star dimension (maximal dimension of a microset). This exhibits the sharpness of recent Furstenberg-type slicing theorems obtained by Algom (2020) about upper bounds on the dimension of every such slice.

Submitted to the Proceedings of Fractals and Related Fields 4