Equidistribution in measure-preserving actions of semisimple groups : case of SL2(R)SL_2(\mathbb{R})

We prove pointwise convergence for the semi-radial averages on $G=SL_2(\mathbb{R})$ given by $\int_t^{t+1} m_K\ast \delta_{a_{s}}ds$ (and similar variants), acting on $K$-finite $L^p$-functions in a probability-measure-preserving action of the group, for $p > 1$

A horospherical ratio ergodic theorem for actions of free groups

We prove a ratio ergodic theorem for amenable equivalence relations satisfying a strong form of the Besicovich covering property. We then use this result to study general non-singular actions of non-abelian free groups and establish a ratio ergodic theorem for averages along horospheres

Geometric covering arguments and ergodic theorems for free groups

We present a new approach to the proof of ergodic theorems for actions of free groups based on geometric covering and asymptotic invariance arguments. Our approach can be viewed as a direct generalization of the classical geometric covering and asymptotic invariance arguments used in the ergodic theory of amenable groups. We use this approach to generalize the existing maximal and pointwise ergodic theorems for free group actions to a large class of geometric averages which were not accessible by previous techniques. Some applications of our approach to other groups and other problems in ergodic theory are also briefly discussed.Comment: In the second version one section was added, proving the pointwise ergodic theorems in L(1) for an amenable equivalence relation with finite invariant measure which posseses an asymptotically invariant sequence satisfying the doubling condition. This is a more abstract formulation of the argument used in the first version, and applies more broadl

Prime Points in Orbits: Some Instances of the Bourgain-Gamburd-Sarnak Conjecture

We use Vaughan's variation on Vinogradov's three-primes theorem to prove Zariski-density of prime points in several infinite families of hypersurfaces, including level sets of some quadratic forms, the Permanent polynomial, and the defining polynomials of some pre-homogeneous vector spaces. Three of these families are instances of a conjecture by Bourgain, Gamburd and Sarnak regarding prime points in orbits of simple algebraic groups. Our approach is based on the formulation of a general condition on the defining polynomial of a hypersurface, which suffices to guarantee that Zariski-density of prime points is equivalent to the existence of an odd point

The ergodic theory of lattice subgroups

We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or S-algebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measure-preserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Gamma, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Gamma, and the domains where the averages are supported. In particular, we establish the general principle that under these conditions a quantitative mean ergodic theorem for a family of averages gives rise to a quantitative solution of the lattice point counting problem in their supports. We demonstrate the new explicit error terms that we obtain by a variety of examples.Comment: 111 pages, Extended appendix on volumes adde

Counting lattice points

For a locally compact second countable group G and a lattice subgroup Gamma, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that i) G has finite upper local dimension, and the domains satisfy a basic regularity condition, ii) the mean ergodic theorem for the action of G on G/Gamma holds, with a rate of convergence. The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality. In addition, it holds uniformly over families of lattices satisfying a uniform spectral gap. Applications include counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties. The mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze-Stein phenomenon. We prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary step in our proof of quantitative results in counting rational points

von-Neumann and Birkhoff ergodic theorems for negatively curved groups

We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT(-1) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells

Quantitative ergodic theorems and their number-theoretic applications

We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Quantitative ergodic theorems give rise to new and previously inaccessible applications, and we demonstrate the remarkable diversity of such applications by presenting several number-theoretic results. These include, in particular, general uniform error estimates in lattice points counting problems, explicit estimates in sifting problems for almost-prime points on symmetric varieties, bounds for exponents of intrinsic Diophantine approximation, and results on fast distribution of dense orbits on homogeneous spaces

Hyperbolic geometry and pointwise ergodic theorems

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.Comment: Editorial changes only, some references added and typos corrected. No change in the main results and their proof

On Arnold's and Kazhdan's equidistribution problems

We consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite, which implies the existence of an associated limiting distribution, possibly different than the invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, which follows from a quantitative mean ergodic theorem for such actions. In turn, these results imply quantitative ratio ergodic theorems for isometric actions of lattices. This sheds some unexpected light on certain equidistribution problems posed by Arnol'd and also on the equidistribution conjecture for dense subgroups of isometries formulated by Kazhdan. We briefly describe the general problem regarding ergodic theorems for actions of lattices on homogeneous spaces and its solution via the duality principle \cite{GN2}, and give a number of examples to demonstrate our results. Finally, we also prove results on quantitative equidistribution for transitive actions.Comment: Submitte