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 | References | Similar Articles | Additional Information

Abstract: Let be a left amenable semigroup. Say that a subset of is large if there is some left invariant mean on with . A subset of is *density recurrent* if and only if, whenever is a large subset of , there is some such that is large. We show that the set of ultrafilters on , every member of which is density recurrent, is a compact subsemigroup of the Stone-Čech compactification of containing the idempotents of . If is a group, we show that for every nonprincipal ultrafilter on , , where . We obtain combinatorial characterizations of sets which are members of a product of idempotents and of sets which are members of a product of elements of the form for each . We show that has substantial multiplicative structure. We show further that if is a large subset of , then , where the quotient set . For each positive integer , we introduce the notion of a *polynomial -recurrent set* in . (Such sets provide a generalization of the polynomial Szemerédi Theorem.) We show that the ultrafilters, every member of which is a polynomial -recurrent set, are a subsemigroup of containing the additive idempotents and a left ideal of .

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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.