Basically you don't need the Weyl's Equi. theorem, it's enough to use Kronecker's lemma about density.

If you want to use measure theory, then your question follows from any ergodic theorem you would like + the fact that the system is uniquely ergodic.

If you want, you can use the theory of continued fractions to explicitly compute such a return time theore.

In the Furstenberg terminology, you want to say that the topological dynamical system of the rotation by $x$ is uniformly recurrent.

If you add nx to itself about ~ 1/(nx) times, you will end up being roughly close to your original nx (this can be made more precise using continued fractions). It is even almost periodic, you always miss the interval (0,v) by at-most one rotation. Hence the set of return times of the orbit of $0$ under the rotation by $x$ map is so-called syndethic set, means it has bounded gaps.

Basically you don't need the Weyl's Equi. theorem, it's enough to use Kronecker's lemma about density.

If you want to use measure theory, then your question follows from any ergodic theorem you would like + the fact that the system is uniquely ergodic.

If you want, you can use the theory of continued fractions to explicitly compute such a return time rate.

In the Furstenberg terminology, you want to say that the topological dynamical system of the rotation by $x$ is uniformly recurrent.

If you add nx to itself about ~ 1/(nx) times, you will end up being roughly close to your original nx (this can be made more precise using continued fractions). It is even almost periodic, you always miss the interval (0,v) by at-most one rotation. Hence the set of return times of the orbit of $0$ under the rotation by $x$ map is so-called syndethic set, means it has bounded gaps.

Basically you don't need the Weyl's Equi. theorem, it's enough to use Kronecker's lemma about density.

If you want to use measure theory, then your question follows from any ergodic theorem you would like + the fact that the system is uniquely ergodic.

If you want, you can use the theory of continued fractions to explicitly compute such a return time theore.

In the Furstenberg terminology, you want to say that the topological dynamical system of the rotation by $x$ is uniformly recurrent.

Say n satisifies that $0<nx<v$, then if you add nx to itself about ~ 1/(nx) times, you will end up being roughly close to your original nx (this can be made more precise using continued fractions). It is even almost periodic, you always miss the interval (0,v) by at-most one rotation. Hence the set of return times of the orbit of $0$ under the rotation by $x$ map is so-called syndethic set, means it has bounded gaps.

Basically you don't need the Weyl's Equi. theorem, it's enough to use Kronecker's lemma about density.

If you want to use measure theory, then your question follows from any ergodic theorem you would like + the fact that the system is uniquely ergodic.

If you want, you can use the theory of continued fractions to explicitly compute such a return time theore.

In the Furstenberg terminology, you want to say that the topological dynamical system of the rotation by $x$ is uniformly recurrent.

If you add nx to itself about ~ 1/(nx) times, you will end up being roughly close to your original nx (this can be made more precise using continued fractions). It is even almost periodic, you always miss the interval (0,v) by at-most one rotation. Hence the set of return times of the orbit of $0$ under the rotation by $x$ map is so-called syndethic set, means it has bounded gaps.