Subsets of Virtually Nilpotent Groups with the SBM Property
Abstract
We extend Leth's notion of subsets of the integers satisfying the Standard interval measure (SIM) property to the class of virtually nilpotent groups and name the corresponding property the Standard ball measure (SBM) property. In order to do this, we define a natural measure on closed balls in asymptotic cones associated to such groups and show that this measure satisfies the Lebesgue density theorem. We then prove analogs of various properties known to hold for SIM sets in this broader context, occasionally assuming extra properties of the group, such as the small spheres property and the small gaps property.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.10683
 Bibcode:
 2022arXiv220610683B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 Mathematics  Logic