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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.