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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, ITALYE-mail: dinasso@dm.unipi.it
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, USAE-mail: isaac@math.uic.edu
RENLING JIN
Affiliation:
DEPARTMENT OF MATHEMATICS COLLEGE OF CHARLESTON CHARLESTON, SC, 29424, USAE-mail: jinr@cofc.edu
STEVEN LETH
Affiliation:
SCHOOL OF MATHEMATICAL SCIENCES UNIVERSITY OF NORTHERN COLORADO CAMPUS BOX 122, 510 20TH STREET GREELEY, CO 80639, USAE-mail: steven.leth@unco.edu
MARTINO LUPINI
Affiliation:
MATHEMATICS DEPARTMENT CALIFORNIA INSTITUTE OF TECHNOLOGY 1200 E. CALIFORNIA BLVD. MC 253-37 PASADENA, CA 91125, USAE-mail: lupini@caltech.edu
KARL MAHLBURG
Affiliation:
DEPARTMENT OF MATHEMATICS LOUISIANA STATE UNIVERSITY 228 LOCKETT HALL BATON ROUGE, LA 70803, USAE-mail: mahlburg@math.lsu.edu
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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

Beiglböck, M., Bergelson, V., and Fish, A., Sumset phenomenon in countable amenable groups . Advances in Mathematics, vol. 223 (2010), no. 2, pp. 416–432.Google Scholar
Björklund, M., personal communication.Google Scholar
Di Nasso, M., Goldbring, I., Jin, R., Leth, S., Lupini, M., and Mahlburg, K., On a sumset conjecture of Erdős . Canadian Journal of Mathematics, 2014, pp. 1–17.Google Scholar
Di Nasso, M., Goldbring, I., Jin, R., Leth, S., Lupini, M., and Mahlburg, K., High density piecewise syndeticity of sumsets . Advances in Mathematics, vol. 278 (2015), pp. 1–33.Google Scholar
Di Nasso, M. and Lupini, M., Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups . Illinois Journal of Mathematics, vol. 58 (2014), no. 1, pp. 11–25.Google Scholar
Jin, R., The sumset phenomenon . Proceedings of the American Mathematical Society, vol. 130 (2002), no. 3, pp. 855–861.Google Scholar
Jin, R., Introduction of nonstandard methods for number theorists, Integers. Electronic Journal of Combinatorial Number Theory, vol. 8 (2008), no. 2.Google Scholar
Tao, T., An Introduction to Measure Theory, Graduate Studies in Mathematics, vol. 126, American Mathematical Society, Providence, RI, 2011.Google Scholar