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ISSN 1088-6850(e) ISSN 0002-9947(p)

Multiple ergodic averages for three polynomials and applications

Author(s): Nikos Frantzikinakis
Journal: Trans. Amer. Math. Soc. 360 (2008), 5435-5475.
MSC (2000): Primary 37A45; Secondary 37A30, 28D05
Posted: April 25, 2008
MathSciNet review: 2415080
Retrieve article in: PDF

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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

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