Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-07-01T18:48:30.476Z Has data issue: false hasContentIssue false

6 - ERGODIC THEORY AND DIOPHANTINE PROBLEMS

Published online by Cambridge University Press:  05 August 2013

F. Blanchard
Affiliation:
Institut de France, Paris
A. Maass
Affiliation:
Universidad de Chile
A. Nogueira
Affiliation:
Universidade Federal do Rio de Janeiro
Get access

Summary

Vitaly BERGELSON

The Ohio State University, Department of Mathematics

231 West 18th Avenue, Columbus, OH 43210–1174

U.S.A.

Introduction

The topic of these notes is the interplay between ergodic theory, some diophantine problems, and the area of combinatorics called Ramsey Theory.

The first section deals with some classical and well-known diophantine results and their connections with topological and measure-preserving dynamics. Some of the proofs offered in Section 6.2 are very elementary, while some use ergodic-theoretic machinery which is actually much more sophisticated than the results it gives us as applications. This should not in any way discourage the reader since the author's intention was not to produce proofs that are as elementary as possible (see Appendix), but to show how intertwined the different and seemingly remote areas of mathematics may be.

The combinatorial results discussed in Sections 6.3, 6.4, and 6.5 are more recent, but they are as beautiful and, in our opinion, as important as the diophantine facts dealt with in Section 6.2.

It was H. Furstenberg, who, with his publication in 1977 of the ergodic-theoretic proof of Szemerédi's theorem (see Section 6.4), established the link between Ramsey theory and the theory of multiple recurrence. Since then, many open problems of combinatorics and number theory have been solved by the methods of ergodic theory and topological dynamics (see for example, [15], [16], [17], [5], and [6]). As it happens, the developments brought to light many new and intriguing problems which are of interest to both the ergodic theory and combinatorics.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2000

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
×