Diophantine approximation with nonsingular integral transformations

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 $V = \mathbb{R}^n \oplus \dotsm \oplus \mathbb{R}^n$ ($p$ copies). We consider the component-wise action of $\Gamma$ on $V$. Let $x \in V$ be such that $\Gamma_x$ is dense in $V$. We discuss the effectiveness of the approximation of any target point $y \in V$ by the orbit $\lbrace \gamma x \vert \gamma \in \Gamma \rbrace$, in terms of ${\lVert \gamma \rVert}$, and prove in particular that for all $x$ 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 $\rho \lt (n-p)/p$, ${\lVert \gamma \mathbf{x} - \mathbf{y} \rVert} \lt {\lVert \gamma \rVert}^{-\rho}$ for infinitely many $\gamma$.

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The authors acknowledge the support of Région Provence-Alpes-Côte d’Azur through the project APEX Systèmes dynamiques: Probabilités et Approximation Diophantienne PAD, CEFIPRA through the project No. 5801-B and the project MATHAMSUD No. 38889 DCS: Dynamics of Cantor Systems.

Received 27 November 2019

Accepted 23 February 2021

Published 29 August 2022