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 $\ell \subseteq {\mathbb{R}}^2$ not parallel to the major axes, $${\dim }_H(\ell \cap F)\le \max \left\{0,\frac{{\dim }_{H}F}{{\dim }^{\ast }F}\cdot ({\dim }^{\ast }F-1)\right\}$$ and $${\dim }_P(\ell \cap F)\le \max \left\{0,\frac{{\dim }_{P}F}{{\dim }^{\ast }F}\cdot ({\dim }^{\ast }F-1)\right\}$$ where ${\dim }^{\ast }$ 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.