On the Hausdorff and packing measures of slices of dynamically defined sets
Let $1\le m m$. 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 $0<ρ<\frac{1}{2}$ let $C_ρ\subset[0,1]$ be the attractor of the IFS $\{f_{ρ,1},f_{ρ,2}\}$, where $f_{ρ,1}(t)=ρ\cdot t$ and $f_{ρ,2}(t)=ρ\cdot t+1-ρ$ 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 $\mathcal{H}^{s-m}(K\cap V)=0$.
math.DS↗