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

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 ${\mathcal{H}}^{s-m}(K\cap V)=0$.