  Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines a
self-map of the $d$-dimensional torus ${\Bbb T}^d={\mathbb{R}}^d/{\Bbb Z}^d$.
Let $ \{E_n \}_{n \in \mathbb{N}} $ be a sequence of subsets of ${\Bbb T}^d$
and let $W(T,\{E_n \})$ be the set of points $\mathbf{x} \in {\Bbb T}^d$ such
that $T^n(\mathbf{x})\in E_n $ for infinitely many $n\in {\mathbb{N}}$. For a
large class of subsets (namely, those satisfying the so called bounded property
$ ({\boldsymbol{\rm B}}) $ which includes balls, rectangles, and hyperboloids)
we show that the $d$-dimensional Lebesgue measure of the shrinking target set
$W(T,\{E_n \})$ is zero (resp. one) if a natural volume sum converges (resp.
diverges). In fact, we prove a quantitative form of this zero-one criteria that
describes the asymptotic behaviour of the counting function $R(x,N):= \# \big\{
1\le n \le N : T^{n}(x) \in E_n \} $. The counting result makes use of a
general quantitative statement that holds for a large class measure-preserving
dynamical systems (namely, those satisfying the so called summable-mixing
property). We next turn our attention to the Hausdorff dimension of $W(T,\{E_n
\})$. In the case the subsets $E_n$ are balls, rectangles or hyperboloids we
obtain precise formulae for the dimension. These shapes correspond,
respectively, to the simultaneous, weighted and multiplicative theories of
classical Diophantine approximation. The dimension results for balls
generalises those obtained in an earlier paper by Hill and the third-named
author for integer matrices to real matrices. In the final section, we discuss
various problems that stem from the results proved in the paper.

Number Theory