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