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High density piecewise syndeticity of product sets in amenable groups

Di Nasso, Mauro and Goldbring, Isaac and Jin, Renling and Leth, Steven and Lupini, Martino and Mahlburg, Karl (2016) High density piecewise syndeticity of product sets in amenable groups. Journal of Symbolic Logic, 81 (4). pp. 1555-1562. ISSN 0022-4812. doi:10.1017/jsl.2015.75. https://resolver.caltech.edu/CaltechAUTHORS:20170113-130158816

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Abstract

M. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB. When G = ℤ^d, this result was first proven by the current set of authors using completely different techniques.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1017/jsl.2015.75DOIArticle
https://arxiv.org/abs/1505.04701arXivDiscussion Paper
ORCID:
AuthorORCID
Lupini, Martino0000-0003-1588-7057
Additional Information:© 2016 The Association for Symbolic Logic. Received August 7, 2015. Published online: 12 August 2016. The authors were supported in part by the American Institute of Mathematics through its SQuaREs program. I. Goldbring was partially supported by NSF CAREER grant DMS-1349399. M. Lupini was supported by the York University Susan Mann Dissertation Scholarship and by the ERC Starting grant no. 259527 of Goulnara Arzhantseva. K. Mahlburg was supported by NSF Grant DMS-1201435. M. Di Nasso was supported by MIUR-PRIN Grant “Models and sets” and by the University of Pisa Grant PRA 2015/005.
Funders:
Funding AgencyGrant Number
American Institute of MathematicsUNSPECIFIED
NSFDMS-1349399
York UniversityUNSPECIFIED
European Research Council (ERC)259527
NSFDMS-1201435
Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR)UNSPECIFIED
University of PisaPRA 2015/005
Issue or Number:4
DOI:10.1017/jsl.2015.75
Record Number:CaltechAUTHORS:20170113-130158816
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170113-130158816
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:73489
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:19 Jan 2017 23:53
Last Modified:11 Nov 2021 05:17

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