1

It is famous that: if $\alpha $ is an irrational number, then $n\alpha$ is dense in $R/Z$. My question is that: if $\alpha_1,...,\alpha_n$ are $n$ $Q$-linear independent irrationals, whether $$(k\alpha_1,...,k\alpha_n) \ (k=1,2..)$$ are dense in $R^n/Z^n$.

I think it is true and try to use induction. However I do not know how the condition that $\alpha_i$ are linear independent can be used. It is necessary because if they are linear dependent, they will be on a hyperplane so can not be dense.

Any help will be appreciated.

user570040
  • 198

1 Answers1

1

Yes. This is true and is known as Kronecker-Weyl theorem. The proof is similar to the proof of Weyl's equidistribution theorem.

asd
  • 903