Arithmetic conditions on self-similar subsets

Let $E,F\subset\mathbb R^d$ be two self-similar sets. Let $\{\rho_i\}$ and $\{\gamma_j\}$ be the contraction ratios of $E$ and $F$, respectively. Suppose that $E$ satisfies the SSC and $F\subset E$. Under certain circumstances, we prove that there exist non-negative rational numbers $t_{i,j}$ such that $\gamma_j=\prod_i\rho_i^{t_{i,j}}$.