Quantitative behavior of unipotent flows and an effective avoidance principle (Q6622463)

\textit{S. G. Dani} and \textit{G. A. Margulis} [Adv. Sov. Math. 16, 91--137 (1993; Zbl 0814.22003)] developed a `linearization' method as a tool to control how long an orbit under the action of a unipotent subgroup spends close to invariant sub-varieties of a homogeneous space. These ideas contributed to an `avoidance' argument along the following lines: An ergodic invariant measure for the action of a connected unipotent subgroup \(U\) on a homogeneous space \(G/\Gamma\) is constrained to be in one of countably many families by Ratner's measure classification [\textit{M. Ratner}, Ann. Math. (2) 134, No. 3, 545--607 (1991; Zbl 0763.28012)], so in settings where \(U\) itself acts ergodically for Haar measure this is one of them. Any other ergodic measure for \(U\) is supported on a proper homogeneous sub-variety of \(G/\Gamma\) and hence for a given growing collection of unipotent orbits that do not spend too long close to one of this countable family the orbits must equidistribute in \(G/\Gamma\). Linearization is used to establish the latter condition in many settings. \par The focus here is on the linearization method itself; the results here will clearly have many applications in homogeneous dynamics, orbit closure classification and Diophantine analysis. Some of these are given here and some will be developed in subsequent work on quantitative density results for unipotent orbits. The setting considered is an arithmetic lattice \(\Gamma<G\) in the \(S\)-arithmetic context (products of real and \(p\)-adic groups) and the main results give effective bounds on the length of time that unipotent orbits can stay close to homogeneous sub-varieties of \(G/\Gamma\) corresponding to \(\mathbb{Q}\)-subgroups of \(G\).