Random Survivor Sets with Moving Holes in Bedford-McMullen Carpets

Let $2\le m<n$ be integers, and let $K(\mathcal{A})$ be the Bedford–McMullen carpet generated by $\{(x,y)\mapsto ((x+i)\mathrm{/}m,(y+j)\mathrm{/}n):(i,j)\in \mathcal{A}\}$, where $\mathcal{A}$ is a subset of $\{0,1,\dots ,m-1\}\times \{0,1,\dots ,n-1\}$. Write $\mathcal{B}$ for the set of occupied columns, and assume that $\mathrm{\#}\mathcal{B}\ge 2$ and each column contains at least two symbols of $\mathcal{A}$. At each vertex of the column tree ${\mathcal{B}}^{\ast }$, we independently choose one forbidden symbol from $\mathcal{A}$ according to a prescribed distribution. Let $\mathbb{P}$ denote the resulting product measure on ${\mathcal{A}}^{{\mathcal{B}}^{\ast }}$. This gives a random survivor set $K_{\omega }(\mathcal{A})$ with $\omega \in {\mathcal{A}}^{{\mathcal{B}}^{\ast }}$. We show that, for $\mathbb{P}$-a.e. $\omega$, \begin{equation*} \dim_H K_ω($\mathcal{A}$)=\frac{1}{\log m}\log\left(\int \sum_{b\in$\mathcal{B}$}\ell_b(ω)^{\frac{\log m}{\log n}} d$\mathbb{P}$(ω)\right), \end{equation*} where ${\mathrm{\ell }}_b(\omega )$ is the number of surviving symbols in column $b$ at the root.