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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 …

Okay, these are just basic technical tidbits. Unsure if anything here is of research level. $SO(n)$ acts transitively on the sphere of $\mathbb{R}^{n}$ (there's a slight ambiguity about $0$, but I gue …

This thing cannot hold no matter what. As Echo rightfully commented, the expression doesn't even compile when $t=0$. It is true one might temper the integral away from $0$, but that's not what you ask …

Edit 2 - Definitely the set of lattices spanned by their shortest vectors is of positive measure. For each such lattice $\Lambda \in SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z})$, there exists $\epsilon=\eps …

Notice that every $m\in SL_{n}(\mathbb{R})$ can be written as $m=u\cdot q$, where $q$ is an ``opposite horospherical'' (maybe with center), namely it is not getting expanded by the $g_{t}$ action. The …

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 …

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 …

Show that for a Bernoulli system, there exists ergodic (Bernoulli) measures of any given entropy (between 0 and full entropy). Pick such a measure with appropriate entropy as you would like. Recall t …

There's a nice proof by Margulis showing that arithmetic subgroups are indeed lattices using the famous Dani-Margulis non-divergence theorem. Actually if you will investigate Ratner's original formula …