Chromatic sumsets

TL;DR

This paper investigates the structure of sumsets formed from finite integer sets, focusing on elements with multiple representations, and provides explicit descriptions for large parameters.

Contribution

It offers a comprehensive analysis of the structure of sumsets with multiple representations for large coefficients, extending previous understanding of sumset behavior.

Findings

Explicit structure formulas for sumsets with multiple representations

Results valid for sufficiently large coefficient vectors

Enhanced understanding of sumset composition in additive combinatorics

Abstract

Let $\mathbf{A} = (A_1,\ldots, A_q)$ be a $q$-tuple of finite sets of integers. Associated to every $q$-tuple of nonnegative integers $\mathbf{h} = (h_1,\ldots, h_q)$ is the linear form $\mathbf{h}\cdot \mathbf{A} = h_1 A_1 + \cdots + h_qA_q$. The set $(\mathbf{h}\cdot \mathbf{A} )^{(t)}$ consists of all elements of this sumset with at least $t$ representations. The structure of the set $(\mathbf{h}\cdot \mathbf{A} )^{(t)}$ is computed for all sufficiently large $h_i$.