On the Hausdorff and packing measures of slices of dynamically defined sets

TL;DR

This paper investigates the Hausdorff and packing measures of slices of self-similar fractal sets, showing that under certain conditions, these measures are typically zero, revealing properties of fractal intersections with affine subspaces.

Contribution

It provides new results on the measure-theoretic properties of slices of self-similar sets, especially for products of Cantor-like sets, under the strong separation condition.

Findings

Hausdorff and packing measures of slices are typically zero

Results apply to self-similar sets satisfying strong separation condition

Specific examples include products of Cantor sets with particular contraction ratios

Abstract

Let $1\le m<n$ be integers, and let $K\subset\mathbb{R}^{n}$ be a self-similar set satisfying the strong separation condition, and with $\dim K=s>m$. We study the a.s. values of the $s-m$-dimensional Hausdorff and packing measures of $K\cap V$, where $V$ is a typical $n-m$-dimensional affine subspace. For $0<\rho<\frac{1}{2}$ let $C_{\rho}\subset[0,1]$ be the attractor of the IFS $\{f_{\rho,1},f_{\rho,2}\}$, where $f_{\rho,1}(t)=\rho\cdot t$ and $f_{\rho,2}(t)=\rho\cdot t+1-\rho$ for each $t\in\mathbb{R}$. We show that for certain numbers $0<a,b<\frac{1}{2}$, for instance $a=\frac{1}{4}$ and $b=\frac{1}{3}$, if $K=C_{a}\times C_{b}$ then typically we have $\mathcal{H}^{s-m}(K\cap V)=0$.