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Transactions of the American Mathematical Society

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Recurrence and primitivity for IP systems with polynomial wildcards


Authors: James T. Campbell and Randall McCutcheon
Journal: Trans. Amer. Math. Soc. 368 (2016), 2697-2721
MSC (2010): Primary 28D05
DOI: https://doi.org/10.1090/tran/6408
Published electronically: May 5, 2015
MathSciNet review: 3449254
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Abstract: The IP Szemerédi Theorem of Furstenberg and Katznelson guarantees that for any positive density subset $ E$ of a countable abelian group $ G$ and for any sequences $ (g_i^{(j)})_{i=1}^\infty $ in $ G$, $ 1\leq j\leq k$, there is a finite non-empty $ \alpha \subset {\mathbb{N}}$ such that $ \bigcap _{j=1}^k ( E- \sum _{i\in \alpha } g_i^{(j)}) \neq \emptyset $. A natural question is whether, in this theorem, one may restrict $ \vert\alpha \vert$ to, for example, the set $ \{ n^d: d\in {\mathbb{N}}\}$. As a first step toward achieving this result, we develop here a new method for taking weak IP limits and prove a relevant projection theorem for unitary operators, which establishes as a corollary the case $ k=2$ of the target result.


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Additional Information

James T. Campbell
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
Email: jcampbll@memphis.edu

Randall McCutcheon
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
Email: rmcctchn@memphis.edu

DOI: https://doi.org/10.1090/tran/6408
Received by editor(s): December 31, 2013
Received by editor(s) in revised form: February 8, 2014, and February 16, 2014
Published electronically: May 5, 2015
Article copyright: © Copyright 2015 American Mathematical Society