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

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>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.</p></div>