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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quotient sets and density recurrent sets


Authors: Vitaly Bergelson and Neil Hindman
Journal: Trans. Amer. Math. Soc. 364 (2012), 4495-4531
MSC (2010): Primary 22A15, 03E05, 05D10; Secondary 54D35
DOI: https://doi.org/10.1090/S0002-9947-2012-05417-1
Published electronically: March 12, 2012
MathSciNet review: 2922599
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Abstract: Let $ S$ be a left amenable semigroup. Say that a subset $ A$ of $ S$ is large if there is some left invariant mean $ \mu $ on $ S$ with $ \mu (\chi _A)>0$. A subset $ B$ of $ S$ is density recurrent if and only if, whenever $ A$ is a large subset of $ S$, there is some $ x\in B$ such that $ x^{-1}A\cap A$ is large. We show that the set $ \mathcal {DR}(S)$ of ultrafilters on $ S$, every member of which is density recurrent, is a compact subsemigroup of the Stone-Čech compactification $ \beta S$ of $ S$ containing the idempotents of $ \beta S$. If $ S$ is a group, we show that for every nonprincipal ultrafilter $ p$ on $ S$, $ p^{-1} p\in \mathcal {DR}(S)$, where $ p^{-1}=\{A^{-1}:A\in p\}$. We obtain combinatorial characterizations of sets which are members of a product of $ k$ idempotents and of sets which are members of a product of $ k$ elements of the form $ p^{-1} p$ for each $ k\in \mathbb{N}$. We show that $ \mathcal {DR}(\mathbb{N},+)$ has substantial multiplicative structure. We show further that if $ A$ is a large subset of $ S$, then $ \mathcal {DR}(S)\subseteq \overline {AA^{-1}}$, where the quotient set $ AA^{-1}=\{x\in S:(\exists y\in A)(xy\in A)\}$. For each positive integer $ n$, we introduce the notion of a polynomial $ n$-recurrent set in $ \mathbb{N}$. (Such sets provide a generalization of the polynomial Szemerédi Theorem.) We show that the ultrafilters, every member of which is a polynomial $ n$-recurrent set, are a subsemigroup of $ (\beta \mathbb{N},+)$ containing the additive idempotents and a left ideal of $ (\beta \mathbb{N},\cdot )$.


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Additional Information

Vitaly Bergelson
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email: vitaly@math.ohio-state.edu

Neil Hindman
Affiliation: Department of Mathematics, Howard University, Washington, DC 20059
Email: nhindman@aol.com

DOI: https://doi.org/10.1090/S0002-9947-2012-05417-1
Received by editor(s): January 2, 2010
Received by editor(s) in revised form: July 7, 2010
Published electronically: March 12, 2012
Additional Notes: The authors acknowledge support received from the National Science Foundation via Grants DMS-0901106 and DMS-0852512 respectively.
Dedicated: Dedicated to Dona Strauss on the occasion of her $75^{th}$ birthday.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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