Triangular and tetrahedral number differences of sumset sizes in additive number theory

TL;DR

This paper explores the distribution of sumset sizes in finite sets of integers, revealing a surprising pattern connected to triangular and tetrahedral numbers, especially for sets of size four.

Contribution

It uncovers a novel pattern in sumset size distributions linked to geometric number sequences, expanding understanding beyond small and large sumset cases.

Findings

Identifies a pattern related to triangular and tetrahedral numbers in sumset sizes.

Highlights the distribution pattern for sets of size four.

Provides insights into the full range of sumset sizes in additive number theory.

Abstract

The study of sums of finite sets of integers has mostly concentrated on sets with very small sumsets (Freiman's theorem and related work) and on sets with very large sumsets (Sidon sets and $B_h$-sets). This paper considers the full range of sumset sizes of finite sets of integers and an unexpected pattern (related to the triangular and tetrahedral numbers) that appears in the distribution of popular sumset sizes of sets of size 4.