High density piecewise syndeticity of product sets in amenable groups

2015/05/18 by Di Nasso, Mauro, Goldbring, Isaac, Jin, Renling +3
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.1505.04701

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.

Related

• Sumset Phenomenon in Countable Amenable Groups
• Product set phenomena for measured groups
• Sumsets of dense sets and sparse sets
• Discrete sumsets with one large summand
• A characterization of piecewise F-syndetic sets