Diophantine approximation with nonsingular integral transformations.

Let $Gamma$ be the multiplicative semigroup of all $ntimes n$ matrices with integral entries and positive determinant. Let $1leq p leq n-1$ and $V=R^noplus cdots oplus R^n$ ($p$ copies). We consider the componentwise action of $Gamma$ on $V$. Let $bxin V$ be such that $Gamma bx$ is dense $V$. We discuss the effectiveness of the approximation of any target point $by in V$ the orbit ${ gamma bx mid gamma in Gamma}$, terms of $norm gamma norm$, and prove particular that for all $bx$ the complement of a specific null set described terms of a certain Diophantine condition, the exponent of approximation is $(n-p)/p$; that is, for any $rhou003c(n-p)/p$, $norm gamma bx - by norm u003c norm gamma norm^{-rho}$ for infinitely many $gamma$.