On Nathanson's Triangular Number Phenomenon

TL;DR

This paper links the early growth of sumset sizes of finite integer sets to lattice minima, explaining the appearance of triangular numbers through combinatorial formulas and geometric interpretations.

Contribution

It introduces a geometric framework connecting sumset sizes to lattice minima, providing a new explanation for the occurrence of triangular numbers in sumset sequences.

Findings

Sumset sizes follow binomial coefficient formulas based on lattice minima.

Early sumset sizes are characterized by simple combinatorial expressions.

Triangular numbers emerge naturally from the geometric interpretation of sumsets.

Abstract

For a finite set $A\subseteq \mathbb{Z}$, the $h$-fold sumset is $hA :=\{x_1+\dots+x_h:x_i\in A\}$. We interpret the beginning of the sequence of sumset sizes $(|hA|)_{h=1}^\infty$ in terms of the successive $L^1$-minima of a lattice (specifically, the points in $\mathbb{Z}^{|A|}$ whose coordinates sum to 0 and which are perpendicular to $\langle a_1,\dots,a_{|A|}\rangle$). In particular, if $h_1,h_2$ are the first and second minima, and $1\le h<h_1$, then $|hA|=\binom{h+|A|-1}{|A|-1}$, while if $h_1\le h <h_2$, then $|hA|=\binom{h+|A|-1}{|A|-1}-\binom{h-h_1+|A|-1}{|A|-1}$. This explains the appearance of triangular numbers in the sequence of sumset sizes, an observation related to a recent experiment of Nathanson.