On Nathanson's Triangular Number Phenomenon

For a finite set A⊆ Z, the h-fold sumset is hA :=₁++xₕ:xᵢ∈ A\. We interpret the beginning of the sequence of sumset sizes (|hA|)ₕ₌₁^∞ in terms of the successive L¹-minima of a lattice (specifically, the points in Z|A| whose coordinates sum to 0 and which are perpendicular to a₁,,a|A|). In particular, if h₁,h₂ are the first and second minima, and 1≤ h<h₁, then |hA|=h+|A|-1|A|-1, while if h₁≤ h <h₂, then |hA|=h+|A|-1|A|-1-h-h₁+|A|-1|A|-1. This explains the appearance of triangular numbers in the sequence of sumset sizes, an observation related to a recent experiment of Nathanson.