# 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 $$i=1,\ldots,n$$.

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.