We present a new approach to metric Diophantine approximation on manifolds based on the
correspondence between approximation properties of numbers and orbit properties of …

Let Q be an indefinite nondegenerate quadratic form in n variables. Let LQ= Q (Zn) denote
the set of values of Q at integral points. The Oppenheim conjecture, proved by GA Margulis …

We introduce a" geometric" method to bound periods of automorphic forms. The key features
of this method are the use of equidistribution results in place of mean value theorems, and …

We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial
sequence of translations on X is well distributed on the union of several sub-nilmanifolds of …

Nilsystems play a key role in the structure theory of measure preserving systems, arising as
the natural objects that describe the behavior of multiple ergodic averages. This book is a …

Unipotent Flows and Counting Lattice Points on Homogeneous Varieties Page 1 Annals of
Mathematics, 143 (1996), 253-299 Unipotent flows and counting lattice points on homogeneous …

Let L be a Lie group and λ a lattice in L. Suppose G is a non-compact simple Lie group
realized as a Lie subgroup of L and GA= L. Let aεG be such that Ada is semisimple and not …

Publisher Summary This chapter presents an exposition of homogeneous dynamics—that is,
the dynamical and ergodic properties of actions on the homogeneous spaces of Lie groups …

A generalized polynomial is a real-valued function which is obtained from conventional
polynomials by the use of the operations of addition, multiplication, and taking the integer …

In this paper we discuss the use of dynamical and ergodic-theoretic ideas and methods to
solve some long-standing problems originating from Lie groups and number theory. These …