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Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces.

Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities

$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^n < T$$

have a nontrivial integer solutions for all large enough $T$. $p\in \mathbb Z^m, q \in \mathbb Z^n$

By Dirichlet approximation, $D_{1,1}=\mathbb R$. I wonder what happens if $(m,n) \ne (1,1)$. Does $D_{m,n}$ necessarily have full measure? I am not sure if there is an elementary proof or ergodic theory/dynamics (things like Dani's correspondence) is needed.

Source of question: the last paragraph of

https://arxiv.org/pdf/1709.04082.pdf#page=2#

Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces.

Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities

$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^n < T$$

have a nontrivial integer solutions for all large enough $T$. $p\in \mathbb Z^m, q \in \mathbb Z^n$

By Dirichlet approximation, $D_{1,1}=\mathbb R$. I wonder what happens if $(m,n) \ne (1,1)$. Does $D_{m,n}$ necessarily have full measure? I am not sure if there is an elementary proof or ergodic theory/dynamics (things like Dani's correspondence) is needed.

Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces.

Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities

$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^n < T$$

have a nontrivial integer solutions for all large enough $T$. $p\in \mathbb Z^m, q \in \mathbb Z^n$

By Dirichlet approximation, $D_{1,1}=\mathbb R$. I wonder what happens if $(m,n) \ne (1,1)$. Does $D_{m,n}$ necessarily have full measure? I am not sure if there is an elementary proof or ergodic theory/dynamics (things like Dani's correspondence) is needed.

Source of question: the last paragraph of

https://arxiv.org/pdf/1709.04082.pdf#page=2#

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No One
No One
  • 1.6k
  • 11
  • 21

Almost every $m\times n$ real matrix is Dirichlet approximable

Let $\| \cdot \|$ denote the maximum norm in Euclidean spaces.

Consider the set $D_{m,n}$ of $m \times n $ real matrices satisfying that the system of inequalities

$$\|Aq-p\|^m < \frac{1}{T}, \|q\|^n < T$$

have a nontrivial integer solutions for all large enough $T$. $p\in \mathbb Z^m, q \in \mathbb Z^n$

By Dirichlet approximation, $D_{1,1}=\mathbb R$. I wonder what happens if $(m,n) \ne (1,1)$. Does $D_{m,n}$ necessarily have full measure? I am not sure if there is an elementary proof or ergodic theory/dynamics (things like Dani's correspondence) is needed.