Diophantine approximation with nonsingular integral transformations

Abstract

Let $\Gamma$ 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 $\Gamma$ on $V$. Let $\bx\in V$ be such that $Γ\bx$ is dense in $V$. We discuss the effectiveness of the approximation of any target point $\by \in V$ by the orbit $\{ γ\bx \mid γ\in Γ\}$, in terms of $\norm γ\norm$, and prove in particular that for all $\bx$ in the complement of a specific null set described in terms of a certain Diophantine condition, the exponent of approximation is $(n-p)/p$; that is, for any $\rho <(n-p)/p$, $\norm γ\bx - \by \norm < \norm γ\norm^{-ρ}$ for infinitely many $\gamma$.

15 pages, proof of Theorem 5.1 corrected