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

TL;DR

This paper establishes improved bounds on the Hausdorff and packing dimensions of line intersections with Bedford-McMullen carpets, advancing Furstenberg-type slicing theorems for self-affine fractals.

Contribution

It provides new upper bounds for dimensions of line slices of Bedford-McMullen carpets, refining previous results for affine invariant fractals.

Findings

Bounds on Hausdorff dimension of line intersections are established.

Bounds on packing dimension of line intersections are established.

Results improve existing Furstenberg 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, $$ \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ and $$ \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ where $\dim^*$ 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.