Subsets of Virtually Nilpotent Groups with the SBM Property

TL;DR

This paper generalizes the SIM property from integers to virtually nilpotent groups by introducing the SBM property, defining measures on asymptotic cones, and establishing key measure-theoretic properties.

Contribution

It extends the SIM property to virtually nilpotent groups through the SBM property and develops a measure theory framework on asymptotic cones.

Findings

Defined a measure on asymptotic cones of virtually nilpotent groups

Proved the Lebesgue density theorem for the SBM property

Established properties analogous to SIM sets in this broader context

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.