JMD
Values of random polynomials at integer points
Jayadev S. Athreya Gregory A. Margulis
Journal of Modern Dynamics 2018, 12(1): 9-16 doi: 10.3934/jmd.2018002

Using classical results of Rogers [12, Theorem 1] bounding the L2-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantitative Oppenheim theorem of Eskin-Margulis-Mozes [6] for almost every quadratic form. Further applications yield quantitative information on the distribution of values of random polynomials at integral points.

keywords: Random lattices Rogers' formula Oppenheim's conjecture
JMD
Logarithm laws for unipotent flows, Ⅱ
Jayadev S. Athreya Gregory A. Margulis
Journal of Modern Dynamics 2017, 11(1): 1-16 doi: 10.3934/jmd.2017001

We prove analogs of the logarithm laws of Sullivan and KleinbockMargulis in the context of unipotent flows. In particular, we prove results for horospherical actions on homogeneous spaces G/Γ.

keywords: Logarithm law unipotent flow norm-like pseudometric
JMD
Logarithm laws for unipotent flows, I
Jayadev S. Athreya Gregory A. Margulis
Journal of Modern Dynamics 2009, 3(3): 359-378 doi: 10.3934/jmd.2009.3.359
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.
keywords: Logarithm laws geometry of numbers. unipotent flows diophantine approximation

Year of publication

Related Authors

Related Keywords

[Back to Top]