Ubiquitous systems and metric number theory

Abstract

We investigate the size and large intersection properties of $$E_t=\{x\in {\mathbb{R}}^d|\Vert x-k-x_i\Vert <{r_i}^t\text{for infinitely many}(i,k)\in I^{\mu ,\alpha }\times {\mathbb{Z}}^d\},$$ 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 $I^{\mu ,\alpha }$ denotes the set of all $i\in I$ such that the $\mu$-mass of the ball with center $x_i$ and radius $r_i$ behaves as ${r_i}^{\alpha }$ for a given Borel measure $\mu$ and a given $\alpha >0$. We establish that the set $E_t$ belongs to the class $\grint^h(\R^d)$ 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.

23 pages

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