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Added some references giving partial results
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Jan-Willem van Ittersum
Jan-Willem van Ittersum
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Kronecker's theorem on diophantine approximation gives a criterium whether the orbit of a tupple ${\underline\alpha}=(\alpha_1,\ldots,\alpha_n)\in \mathbb{R}^n/\mathbb{Z}^n$ comes arbitrarily close to another tuple $\underline x$ under the diagonal action of $a\in\mathbb{Z}$ given by multiplication by $a$, that is whether there exist $a\in \mathbb{Z}$ and $\underline k\in \mathbb{Z}^n$ such that for all $\epsilon>0$ $$|a\alpha_i-x_i-k_i|<\epsilon$$ for all $i=1,\ldots,n$.

Question Now, replace $\mathbb{R}$ by a lattice $\mathbb{R}^2$ and the diagonal action of $\mathbb{Z}$ by the action of $\mathrm{SL}_2(\mathbb{Z})$. What is the generalisation of Kronecker's theorem in this setting? So, given ${\underline\alpha},{\underline\beta}\in \mathbb{R}^n/\mathbb{Z}^n$ and $\underline x,\underline y\in \mathbb{R}^n/\mathbb{Z}^n$, does there exist $\left( \begin{smallmatrix} a& b\\ c & d \end{smallmatrix}\right)\in \mathrm{SL}_2(\mathbb{Z})$$\gamma=\left( \begin{smallmatrix} a& b\\ c & d \end{smallmatrix}\right)\in \mathrm{SL}_2(\mathbb{Z})$ and $\underline k, \underline l\in \mathbb{Z}^n$ such that for all $\epsilon>0$ one has $$|a\alpha_i+b\beta_i-x_i-k_i|<\epsilon \quad \text{and} \quad |c\alpha_i+d\beta_i-y_i-l_i|<\epsilon$$ for all $i=1,\ldots,n$?

References The paper Diophantine approximation exponents on homogeneous varieties in Section 4.3 gives estimates for the size of the matrix $\gamma$, assuming $\gamma$ exists. In the case $n=1$, a statement which can be found in for example Approximation to points in the plane by SL (2, Z)-orbits says that in case $n=1$ and the ratio of $\alpha_1$ and $\beta_1$ is irrationial, the orbit of $(\alpha,\beta)$ under $\mathrm{SL}_2(\mathbb{Z})$ is dense, hence giving an affirmative answer to my question.

Kronecker's theorem on diophantine approximation gives a criterium whether the orbit of a tupple ${\underline\alpha}=(\alpha_1,\ldots,\alpha_n)\in \mathbb{R}^n/\mathbb{Z}^n$ comes arbitrarily close to another tuple $\underline x$ under the diagonal action of $a\in\mathbb{Z}$ given by multiplication by $a$, that is whether there exist $a\in \mathbb{Z}$ and $\underline k\in \mathbb{Z}^n$ such that for all $\epsilon>0$ $$|a\alpha_i-x_i-k_i|<\epsilon$$ for all $i=1,\ldots,n$.

Now, replace $\mathbb{R}$ by a lattice $\mathbb{R}^2$ and the diagonal action of $\mathbb{Z}$ by the action of $\mathrm{SL}_2(\mathbb{Z})$. What is the generalisation of Kronecker's theorem in this setting? So, given ${\underline\alpha},{\underline\beta}\in \mathbb{R}^n/\mathbb{Z}^n$ and $\underline x,\underline y\in \mathbb{R}^n/\mathbb{Z}^n$, does there exist $\left( \begin{smallmatrix} a& b\\ c & d \end{smallmatrix}\right)\in \mathrm{SL}_2(\mathbb{Z})$ and $\underline k, \underline l\in \mathbb{Z}^n$ such that for all $\epsilon>0$ one has $$|a\alpha_i+b\beta_i-x_i-k_i|<\epsilon \quad \text{and} \quad |c\alpha_i+d\beta_i-y_i-l_i|<\epsilon$$ for all $i=1,\ldots,n$?

Kronecker's theorem on diophantine approximation gives a criterium whether the orbit of a tupple ${\underline\alpha}=(\alpha_1,\ldots,\alpha_n)\in \mathbb{R}^n/\mathbb{Z}^n$ comes arbitrarily close to another tuple $\underline x$ under the diagonal action of $a\in\mathbb{Z}$ given by multiplication by $a$, that is whether there exist $a\in \mathbb{Z}$ and $\underline k\in \mathbb{Z}^n$ such that for all $\epsilon>0$ $$|a\alpha_i-x_i-k_i|<\epsilon$$ for all $i=1,\ldots,n$.

Question Now, replace $\mathbb{R}$ by $\mathbb{R}^2$ and the diagonal action of $\mathbb{Z}$ by the action of $\mathrm{SL}_2(\mathbb{Z})$. What is the generalisation of Kronecker's theorem in this setting? So, given ${\underline\alpha},{\underline\beta}\in \mathbb{R}^n/\mathbb{Z}^n$ and $\underline x,\underline y\in \mathbb{R}^n/\mathbb{Z}^n$, does there exist $\gamma=\left( \begin{smallmatrix} a& b\\ c & d \end{smallmatrix}\right)\in \mathrm{SL}_2(\mathbb{Z})$ and $\underline k, \underline l\in \mathbb{Z}^n$ such that for all $\epsilon>0$ one has $$|a\alpha_i+b\beta_i-x_i-k_i|<\epsilon \quad \text{and} \quad |c\alpha_i+d\beta_i-y_i-l_i|<\epsilon$$ for all $i=1,\ldots,n$?

References The paper Diophantine approximation exponents on homogeneous varieties in Section 4.3 gives estimates for the size of the matrix $\gamma$, assuming $\gamma$ exists. In the case $n=1$, a statement which can be found in for example Approximation to points in the plane by SL (2, Z)-orbits says that in case $n=1$ and the ratio of $\alpha_1$ and $\beta_1$ is irrationial, the orbit of $(\alpha,\beta)$ under $\mathrm{SL}_2(\mathbb{Z})$ is dense, hence giving an affirmative answer to my question.

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Jan-Willem van Ittersum
Jan-Willem van Ittersum
  • 399
  • 1
  • 10

Kronecker's theorem on diophantine approximation for $\mathrm{SL}_2(\mathbb{Z})$

Kronecker's theorem on diophantine approximation gives a criterium whether the orbit of a tupple ${\underline\alpha}=(\alpha_1,\ldots,\alpha_n)\in \mathbb{R}^n/\mathbb{Z}^n$ comes arbitrarily close to another tuple $\underline x$ under the diagonal action of $a\in\mathbb{Z}$ given by multiplication by $a$, that is whether there exist $a\in \mathbb{Z}$ and $\underline k\in \mathbb{Z}^n$ such that for all $\epsilon>0$ $$|a\alpha_i-x_i-k_i|<\epsilon$$ for all $i=1,\ldots,n$.

Now, replace $\mathbb{R}$ by a lattice $\mathbb{R}^2$ and the diagonal action of $\mathbb{Z}$ by the action of $\mathrm{SL}_2(\mathbb{Z})$. What is the generalisation of Kronecker's theorem in this setting? So, given ${\underline\alpha},{\underline\beta}\in \mathbb{R}^n/\mathbb{Z}^n$ and $\underline x,\underline y\in \mathbb{R}^n/\mathbb{Z}^n$, does there exist $\left( \begin{smallmatrix} a& b\\ c & d \end{smallmatrix}\right)\in \mathrm{SL}_2(\mathbb{Z})$ and $\underline k, \underline l\in \mathbb{Z}^n$ such that for all $\epsilon>0$ one has $$|a\alpha_i+b\beta_i-x_i-k_i|<\epsilon \quad \text{and} \quad |c\alpha_i+d\beta_i-y_i-l_i|<\epsilon$$ for all $i=1,\ldots,n$?