Carleson families of cubes related to porous sets

Given a porous set $E\in {\mathbb{R}}^d$ and a dyadic lattice $\mathcal{D}$, we refine the Carleson packing condition and the sparseness property for the dyadic cover ${\mathcal{D}}_E=\{Q\in \mathcal{D}:Q\cap E\mathrm{\ne }\mathrm{\varnothing }\}$. We study the inverse problem, when a Carleson family $\mathcal{S}\subset \mathcal{D}$ generates the porous set $E$ such that $\mathcal{S}\subset {\mathcal{D}}_E$.