Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Multiple ergodic averages for three polynomials and applications

Author: Nikos Frantzikinakis
Journal: Trans. Amer. Math. Soc. 360 (2008), 5435-5475
MSC (2000): Primary 37A45; Secondary 37A30, 28D05
Published electronically: April 25, 2008
MathSciNet review: 2415080
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form $ \{l_1p,l_2p,\ldots,l_kp\}$. We then derive several multiple recurrence results and combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all $ \varepsilon>0$ and every subset of the integers $ \Lambda$ the set

$\displaystyle \big\{n\in\mathbb{N}\colon d^*\big(\Lambda\cap (\Lambda+p_1(n))\cap (\Lambda+p_2(n))\cap (\Lambda+ p_3(n))\big)>(d^*(\Lambda))^4-\varepsilon\big\} $

has bounded gaps for ``most'' choices of integer polynomials $ p_1,p_2,p_3$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37A45, 37A30, 28D05

Retrieve articles in all journals with MSC (2000): 37A45, 37A30, 28D05

Additional Information

Nikos Frantzikinakis
Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152-3240

PII: S 0002-9947(08)04591-1
Keywords: Characteristic factor, multiple ergodic averages, multiple recurrence, polynomial Szemer\'edi.
Received by editor(s): October 17, 2006
Published electronically: April 25, 2008
Additional Notes: The author was partially supported by NSF grant DMS-0111298.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia