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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
DOI: https://doi.org/10.1090/S0002-9947-08-04591-1
Published electronically: April 25, 2008
MathSciNet review: 2415080
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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$.


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Additional Information

Nikos Frantzikinakis
Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152-3240
Email: frantzikinakis@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-08-04591-1
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.

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