Search Results

It is unclear what is "most $r$'s even mean. A standard argument would show that for any increasing sequence, for Lebesgue almost every $x$, $a_{n}.x$ is equidistributed mod $1$. For the case of power …

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 …

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 …

Here's one example that I like. Consider $\Gamma = \{g \in SL_d\left[\sqrt{-m} / p\right] \mid g^t \cdot g^\sigma= I \}$, where $\sigma$ is the Galois conjugate. Then this is an arithmetic lattice in …

It is a whole line of ideas (and proofs) which go usually by the name of ``Linnik's problems''. Apart from Linnik's book (and the Linnik-Skubenko theorem), it has been extensively studied by many rese …

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 …

Ok, let take $v$ in the bundle at $x$. We may decompose $v=v^{u}+v^{s}\in E^{u}\oplus E^{s}$. Assume without loss of generality that $\lVert v\rVert=1$. Applying $A^{i}$, using equivariance and Osceld …

$\DeclareMathOperator\supp{supp}$The theorem holds for semigroups as well (well, in the finite volume setting! in the infinite volume setting there are subtleties between two-sided and one-sided avera …

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 …

This observation is attributed to H. Furstenberg, and appears (in the case of shift-invariant sets, i.e. Cantor sets) in his beautiful Disjointness paper (in section $3$, which you can read independen …

EDIT - The answer below deals with an ergodic m.p.s As this question got up-voted, I've decided to fuly write a solution, based on the sketch I've made in the comments. Fix some $\varepsilon>0$ smal …

In general, it varies. There are cases where the convergence is quite fast (for example in the case where the system is mixing, and say in the presence of spectral gap, think of Bernoulli system or sa …

This is indeed true for some "nice systems", for example one can show this theorem (for say $L^{2}$-functions) for Kronecker systems simply by van-der-Corput trick. In general, those averages converg …

This relatively straight forward. The main observation is that, in the Host Meiri terminology, the $p^{N}$-cells are exactly the inverse image of the $p^{N-1}$-cells, and the $\times p$ map is $p$ to …

Well that will be some lengthy answer. The first article that was published after the famous disjointness paper is another paper by Hillel called "Intersections of Cantor sets", it's related to the m …