Diophantine approximation with nonsingular integral transformations

Let $\mathrm{\Gamma }$ be the multiplicative semigroup of all $n\times n$ matrices with integral entries and positive determinant. Let $1\le p\le n-1$ and $V={\mathbb{R}}^n\oplus \cdots \oplus {\mathbb{R}}^n$ ($p$ copies). We consider the componentwise action of $\mathrm{\Gamma }$ on $V$. Let $\text{bx}\in V$ be such that $\mathrm{\Gamma }\text{bx}$ is dense in $V$. We discuss the effectiveness of the approximation of any target point $\by \in V$bytheorbit${ \gamma \bx \mid \gamma \in \Gamma}$,intermsof$\norm \gamma \norm$,andproveinparticularthatforall$\bx$ in the complement of a specific null set described in terms of a certain Diophantine condition, the exponent of approximation is $(n-p)\mathrm{/}p$; that is, for any $\rho <(n-p)\mathrm{/}p$, $\text{norm}\gamma \text{bx}-\text{by}\text{norm}<\text{norm}\gamma {\text{norm}}^{-\rho }$ for infinitely many $\gamma$.