Today when I was doing ergodic theory problems I faced with following problem:
Assume rotation map on $k$-dimensional torus under $\alpha=(\alpha_1,...,\alpha_n)$ then orbit of all $x$ in $k$-dimensional torus is dense if and only if vector $\alpha$ is irrationally independent
( means $n_o+n_1\alpha_1+...+n_n\alpha_n=0 \quad \text{then}\quad n_i=0 \quad \text{for all}\quad n_i \in \mathbb Z$ )
I solved density of orbit of irrational rotation on circle but when i generalized it to torus problem couldn't be solved by that technique. on the other side I tried to translate trajectory on $2$ dim torus to unit square which trajectory starts from zero but how to find angle of trajectory associated to rotation.
"Some Questions has been asked but their answers wasn't clear for me? I will be appreciate if add some hint.