Logarithm laws for unipotent flows, I

Abstract

We prove analogs of the logarithm laws of Sullivan and Kleinbock-Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices SL(n,R)/ SL(n,Z). The key lemma for our results says the measure of the set of unimodular lattices in R n that does not intersect a 'large' volume subset of R n is 'small'. This can be considered as a 'random' analog of the classical Minkowski Theorem in the geometry of numbers.

Shrinking targets for semisimple groups

“…For non‐diagonalizable actions, less is known. Here Athreya and Margulis considered the cases where is the expanding horospherical group corresponding to a diagonalizable flow as well as the case where is a 1‐dimensional unipotent group acting on the space of lattices , and in both cases they managed to prove logarithm laws. Similar results were also obtained by Kelmer and Mohammadi for cusp excursions of one parameter unipotent flows on spaces of the form , where is a product of a number of copies of and and is irreducible, and more recently in a forthcoming paper by Yu when .…”

Khintchine-type theorems for values of subhomogeneous functions at integer points

“…Let B denote the closed Euclidean ball in R n that is centered at the origin and whose measure is equal to m(E). By Lemma 4.2 in [AM09] and the proof of Theorem 2.2 in [AM09] (see Section 4.1 of [AM09]), it follows…”

Shrinking targets on square-tiled surfaces

“…Here, the authors explicitly use a shrinking target argument. In 2009, Athreya and Margulis [2] [3] extended Sullivan's result to unipotent flows on the space of lattices also using a shrinking target argument. For the interested reader, many of these results are put into a more general framework in a survey by Athreya [1].…”

“…Definition 1.1 (Square-tiled Surface [15] [24]). A square-tiled surface is a pair (X, ω), where X is a closed Riemann surface and ω a holomorphic 1-form, given by a (finite) branched cover over the square torus, π : X → T 2 , branched over 0. The one-form ω is given by the pullback of dz under the covering map π, ω = π * (dz).…”