Ubiquitous systems and metric number theory

We investigate the size and large intersection properties of $$E_t=\{x\in {\mathbb{R}}^d\mathrm{\mid }\mathrm{\parallel }x-k-x_i\mathrm{\parallel }<{r_i}^t\text{for infinitely many}(i,k)\in I^{\mu ,\alpha }\times {\mathbb{Z}}^d\},$$ where $d\in \mathbb{N}$, $t\ge 1$, $I$ is a denumerable set, $(x_i,r_i{)}_{i\in I}$ is a family in $[0,1]^d\times (0,\infty)$and$I^{\mu,\alpha}$denotesthesetofall$i\in I$suchthatthe$\mu$-massoftheballwithcenter$x_{i}$andradius$r_{i}$behavesas${r_{i}}^\alpha$foragivenBorelmeasure$\mu$andagiven$\alpha>0$. We establish that the set $E_t$ belongs to the class ${\text{grint}}^h({\mathbb{R}}^d)$ of sets with large intersection with respect to a certain gauge function $h$, provided that $(x_i,r_i{)}_{i\in I}$ is a heterogeneous ubiquitous system with respect to $\mu$. In particular, $E_t$ has infinite Hausdorff $g$-measure for every gauge function $g$ that increases faster than $h$ in a neighborhood of zero. We also give several applications to metric number theory.