Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T19:09:49.268Z Has data issue: false hasContentIssue false

18 - Diophantine Approximation, Lattices and Flows on Homogeneous Spaces

Published online by Cambridge University Press:  20 August 2009

Gisbert Wüstholz
Affiliation:
Swiss Federal University (ETH), Zürich
Get access

Summary

Introduction

During the last 15–20 years it has been realized that certain problems in Diophantine approximation and number theory can be solved using geometry of the space of lattices and methods from the theory of flows on homogeneous spaces. The purpose of this survey is to demonstrate this approach on several examples. We will start with Diophantine approximation on manifolds where we will briefly describe the proof of Baker–Sprindžuk conjectures and some Khintchine-type theorems. The next topic is the Oppenheim conjecture proved in the mid-1980s and the Littlewood conjecture, still not settled. After that we will go to quantitative generalizations of the Oppenheim conjecture and to counting lattice points on homogeneous varieties. In the last part we will discuss results on unipotent flows on homogeneous spaces which play, directly or indirectly, the most essential role in the solution of the above-mentioned problems. Most of those results on unipotent flows are proved using ergodic theorems and also notions such as minimal sets and invariant measures. These theorems and notions have no effective analogs and because of that the homogeneous space approach is not effective in a certain sense. We will briefly discuss the problem of the effectivization at the very end of the paper.

The author would like to thank A. Eskin and D. Kleinbock for their comments on a preliminary version of this article.

Diophantine approximation on manifolds

We start by recalling some standard notation and terminology.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×