Improved Versions of Some Furstenberg Type Slicing Theorems for Self-Affine Carpets

Abstract

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, $$\begin{align*} & \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace\end{align*}$$and $$\begin{align*} & \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace,\end{align*}$$where $\dim ^*$ is Furstenberg’s star dimension (maximal dimension of microsets). This improves the state-of-the-art results on Furstenberg type slicing Theorems for affine invariant carpets.