Shrinking targets for semisimple groups

Abstract

We study the shrinking target problem for actions of semisimple groups on homogeneous spaces, with applications to logarithm laws and Diophantine approximation related to an effective version of the Oppenheim conjecture valid for almost all quadratic forms.

Shrinking targets for discrete time flows on hyperbolic manifolds

“…Using the ideas of [GK15], by showing that the H action satisfies effective mean ergodic theorems with exponents κ = 1/2 (or arbitrarily close to 1/2), it is possible to essentially resolve the shrinking target problem for the corresponding H action. Our goal in this section is to prove such bounds for the case where X = Γ\G and the group H ≤ G is either a unipotent group (of rank d) or a diagonalizable group (of rank one).…”

Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

“…The use of quantitative mixing of geodesic flow in the shrinking target problem in the homogeneous setting goes back to the work of Kleinbock and Margulis [16], and the idea of using an effective mean ergodic theorem was first introduced in [9] and more explicitly in [13,14], where these ideas were used to prove the analogous results for finite volume hyperbolic manifold. In this paper we follow closely the arguments of [13], in particular, for the discretized geodesic flows; once we establish the relevant result on exponential decay of matrix coefficients as in Theorem 1.14, the arguments are essentially identical, although establishing the needed regularity conditions for explicit examples is technically more difficult in the infinite volume case (see Section 5).…”

Topics in homogeneous dynamics and number theory

“…We will not elaborate on this theme in this survey, referring the reader instead to [GGN5] for details and to [GGN3,GGN4] for the more general study of effective density of lattice orbits on homogeneous varieties and [GGN1,GGN2] for the related problem of intrinsic Diophantine approximation on varieties. See also [GK2] for a related question on quadratic forms studied originally by Bourgain [B].…”