JMD
Logarithm laws for unipotent flows, I
Jayadev S. Athreya Gregory A. Margulis
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
JMD
Logarithm laws for unipotent flows, Ⅱ
Jayadev S. Athreya Gregory A. Margulis

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
Values of random polynomials at integer points
Jayadev S. Athreya Gregory A. Margulis

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

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