Heterogeneous ubiquitous systems in $\mathbb{R}^{d}$ and Hausdorff dimension

Let $\{x_n\}_{n\geq 0}$ be a sequence of $[0,1]^d$, $\{\lambda_n\} _{n\geq 0}$ a sequence of positive real numbers converging to 0, and $\delta>1$. Let $\mu$ be a positive Borel measure on $[0,1]^d$, $\rho\in (0,1]$ and $\alpha>0$. Consider the limsup-set \[S_{\mu}(\rho,\delta,\alpha)= \bigcap_{N\in \mathbb{N}} \bigcup _{n\geq N: \mu(B(x_n,\lambda^\rho_n)) \sim \lambda_n^{\rho\alpha}} B(x_n,\lambda_n^\delta).\] We show that, under suitable assumptions on the measure $\mu$, the Hausdorff dimension of the sets $S_{\mu}(\rho,\delta,\alpha)$ can be computed. When $\rho