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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 …

I'm pretty sure that plenty of those kind of questions are covered in Cassels' book. The modern approach to this kind of problems follows from dynamics on homogeneous spaces via Dani's correspondence …

A number is in BA if its orbit is bounded. Any such orbit closure must contain a full $A=\langle g_t\rangle$ orbit. By examining the possible subgroups, any such hypothetical $H$, as a stability group …

It is evident that the singular vectors are defined as the ``$u_{A}$-part which is $g_{t}$ divergent in the future'', this gives $m\cdot n$ ($=\dim \left(u_{A}\right)$) minus the dimension of the sing …

First of all, $Y$ is not called the “grid space”. It is sometimes called the affine space and can be identified with a quotient of the affine group $\operatorname{ASL}_{n}$, namely the semi-direct pro …

You've received two good answers, but I'll elaborate a bit. Usually equidistribution on the torus (or more general, compact groups) wrt the Haar measure is achieved by computing the Weyl sums and show …