Recurrence and primitivity for IP systems with polynomial wildcards
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- by James T. Campbell and Randall McCutcheon PDF
- Trans. Amer. Math. Soc. 368 (2016), 2697-2721 Request permission
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 $|\alpha |$ 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.References
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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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 2697-2721
- MSC (2010): Primary 28D05
- DOI: https://doi.org/10.1090/tran/6408
- MathSciNet review: 3449254