Diophantine approximation with nonsingular integral transformations
| Autor: | Dani, S. G., Nogueira, Arnaldo |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $\gamma$. Comment: 15 pages, proof of Theorem 5.1 corrected |
| Databáze: | arXiv |
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