Subsets of virtually nilpotent groups with the SBM property

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 property. In order to do this, we define a natural measure on closed balls in asymptotic cones associated with 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.