Ubiquitous systems and metric number theory

We investigate the size and large intersection properties of E-t = {x is an element of R-d vertical bar parallel to x - k - x(i)parallel to <r(i)(t) for infinitely many (i,k) is an element of I-mu,I-alpha x Z(d)}, where d is an element of N, t >= 1, I is a denumerable set, (x(i), r(i))(i is an element of 1) is a family in [0, 1](d) x (0, infinity) and I-mu,I-alpha denotes the set of all i is an element of I such that the mu-mass of the ball with center x(i) and radius r(i) behaves as r(i) for a given Borel measure mu and a given alpha > 0. We establish that the set E-t belongs to the class G(h)(R-d) of sets with large intersection with respect to a certain gauge function h, provided that (x(i), r(i))(i is an element of 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. (c) 2007 Elsevier Inc. All rights reserved.