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HIGH DENSITY PIECEWISE SYNDETICITY OF PRODUCT SETS IN AMENABLE GROUPS

Published online by Cambridge University Press:  12 August 2016

MAURO DI NASSO ,
ISAAC GOLDBRING ,
RENLING JIN ,
STEVEN LETH ,
MARTINO LUPINI  and
KARL MAHLBURG
MAURO DI NASSO
Affiliation:
DIPARTIMENTO DI MATEMATICA UNIVERSITA’ DI PISA LARGO BRUNO PONTECORVO 5 PISA 56127, ITALY E-mail: [email protected]
ISAAC GOLDBRING
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO SCIENCE AND ENGINEERING OFFICES M/C 249 851 S. MORGAN ST., CHICAGO, IL, 60607-7045, USA E-mail: [email protected]
RENLING JIN
Affiliation:
DEPARTMENT OF MATHEMATICS COLLEGE OF CHARLESTON CHARLESTON, SC, 29424, USA E-mail: [email protected]
STEVEN LETH
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES UNIVERSITY OF NORTHERN COLORADO CAMPUS BOX 122, 510 20TH STREET GREELEY, CO 80639, USA E-mail: [email protected]
MARTINO LUPINI
Affiliation:
MATHEMATICS DEPARTMENT CALIFORNIA INSTITUTE OF TECHNOLOGY 1200 E. CALIFORNIA BLVD. MC 253-37 PASADENA, CA 91125, USA E-mail: [email protected]
KARL MAHLBURG
Affiliation:
DEPARTMENT OF MATHEMATICS LOUISIANA STATE UNIVERSITY 228 LOCKETT HALL BATON ROUGE, LA 70803, USA E-mail: [email protected]
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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 = ℤd , this result was first proven by the current set of authors using completely different techniques.

Keywords

11B1311B0503H0543A07amenable groupspiecewise syndeticityproduct sets
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 4 , December 2016 , pp. 1555 - 1562
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

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