Logarithm laws for unipotent flows, I

By Jayadev S. Athreya, Grigorii Margulis. 2008

77 citations recorded by the source corpus.

We prove analogues 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' analogue of the classical Minkowski theorem in the geometry of numbers.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5eaaa20760b36446003c0d7ca95dd32654c63a1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Jayadev S. Athreya
2. Y. Cheung
3. Jens Marklof
4. Artur Avila
5. S. Gouëzel
6. Berk Ustun
7. Cynthia Rudin

In Mathematical Physics

1. Jayadev S. Athreya
2. Jens Marklof
3. Y. Cheung
4. Andreas Strombergsson
5. Artur Avila
6. S. Gouëzel

Citations in the corpus

• A Poincar\'e section for horocycle flow on the space of lattices (2012, background): “Margulis [4], showed that for any Λ ∈ X2 without vertical vectors,”
• Fine-Scale Statistics for the Multidimensional Farey Sequence (2012, background): “It is however possible to provide asymptotic estimates for large and small sets A when k = 0 and k = 1, see [2, 18] for general results in this direction.”