# Diophantine approximation with nonsingular integral transformations

## TL;DR

This paper studies how well points in a high-dimensional space can be approximated by orbits of integer matrices with positive determinant, establishing a specific approximation exponent for most points outside a null set.

## Contribution

It introduces a new Diophantine approximation result for orbits under nonsingular integral matrix transformations, identifying the approximation exponent for generic points.

## Key findings

- Approximation exponent is (n-p)/p for almost all points outside a null set.
- The effectiveness of approximation depends on the Diophantine properties of the initial point.
- The results extend classical Diophantine approximation to matrix orbits in higher dimensions.

## Abstract

Let $\Gamma$ be the multiplicative semigroup of all $n\times n$ matrices with integral entries and positive determinant. Let $1\leq p \leq n-1$ and $V=\R^n\oplus \cdots \oplus \R^n$ ($p$ copies). We consider the componentwise action of $\Gamma$ on $V$. Let $\bx\in V$ be such that $\Gamma \bx$ is dense in $V$. We discuss the effectiveness of the approximation of any target point $\by \in V$ by the orbit $\{ \gamma \bx \mid \gamma \in \Gamma\}$, in terms of $\norm \gamma \norm$, 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 $\rho<(n-p)/p$, $\norm \gamma \bx - \by \norm < \norm \gamma \norm^{-\rho}$ for infinitely many $\gamma$.