  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 $\rho<1$, a yet unknown
saturation phenomenon appears in the computation of the Hausdorff dimension of
$S\_{\mu} (\rho,\delta, \alpha)$. Our results apply to several classes of
multifractal measures $\mu$. The computation of the dimensions of such sets
opens the way to the study of several new objects and phenomena. Applications
are given for the Diophantine approximation conditioned by (or combined with)
$b$-adic expansion properties, by averages of some Birkhoff sums and by
asymptotic behavior of random covering numbers.
