Ergodic Theory on Homogeneous Spaces and Metric Number Theory

Article outline This article gives a brief overview of recent developments in metric number theory, in particular, Diophantine approximation on manifolds, obtained by applying ideas and methods coming from dynamics on homogeneous spaces. Glossary 1. Definition: Metric Diophantine approximation 2. Basic facts 3. Introduction 4. Connection with dynamics on the space of lattices 5. Diophantine approximation with dependent quantities 6. Further results 7. Future directions References Glossary Diophantine approximation Diophantine approximation refers to approximation of real numbers by rational numbers, or more generally, finding integer points at which some (possibly vector-valued) functions attain values close to integers. metric number theory Metric number theory (or, specifically, metric Diophantine approximation) refers to the study of sets of real numbers or vectors with prescribed Diophantine approximation properties. homogeneous spaces A homogeneous space G/Γ of a group G by its subgroup Γ is the space of cosets {gΓ}. When G is a Lie group and Γ is a discrete subgroup, the space G/Γ is a smooth manifold and locally looks like G itself. lattice; unimodular lattice A lattice in a Lie group is a discrete subgroup of finite covolume; unimodular stands for covolume equal to 1.