Ubiquitous systems and metric number theory

We investigate the size and large intersection properties ofEt={x∈Rd|‖x−k−xi‖0α>0. We establish that the set EtEt belongs to the class Gh(Rd)Gh(Rd) of sets with large intersection with respect to a certain gauge function h  , provided that (xi,ri)i∈I(xi,ri)i∈I is a heterogeneous ubiquitous system with respect to μ  . In particular, EtEt 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.