Sums of Finite Sets of Integers, II

TL;DR

This paper investigates the structure of the subset of integers with multiple representations as sums of elements from a finite set, providing a complete characterization for sufficiently large sum counts.

Contribution

It offers a complete description of the structure of the set of integers with at least t representations as sums from a finite set for all large enough h.

Findings

The structure of (hA)^{(t)} is fully characterized for all h ≥ h_t.

Provides conditions under which the structure stabilizes.

Extends previous results on sumsets of finite integer sets.

Abstract

Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a sum of $h$ elements of $\mathcal{A}$. The structure of the set $(h\mathcal{A})^{(t)}$ is completely determined for all $h \geq h_t$.