Direct and Inverse Theorems on Signed Sumsets of Integers

TL;DR

This paper investigates the size and structure of signed sumsets of finite integer sets, providing solutions to both the direct problem of lower bounds and the inverse problem of characterizing sets with minimal sumset size.

Contribution

It offers new results solving the direct and inverse problems for signed sumsets of integers, advancing understanding of their combinatorial structure.

Findings

Established lower bounds for the size of signed sumsets.

Characterized the structure of sets with minimal signed sumsets.

Extended classical sumset results to signed sumsets in integers.

Abstract

Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i} a_{i}: (\lambda_{0}, \ldots, \lambda_{k-1}) \in \mathbb{Z}^{k},~ \Sigma_{i=0}^{k-1}|\lambda_{i}|=h \},\] be the {\it signed sumset} of $A$. The {\it direct problem} for the signed sumset $h_{\underline{+}}A$ is to find a nontrivial lower bound for $|h_{\underline{+}}A|$ in terms of $|A|$. The {\it inverse problem} for $h_{\underline{+}}A$ is to determine the structure of the finite set $A$ for which $|h_{\underline{+}}A|$ is minimal. In this article, we solve both the direct and inverse problems for $|h_{\underline{+}}A|$, when $A$ is a finite set of integers.