Direct and Inverse Theorems on Signed Sumsets of Integers

Abstract

Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0,a_1,\dots ,a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{Σ_{i=0}^{k-1}λ_{i} a_{i}: (λ_{0}, \ldots, λ_{k-1}) \in \mathbb{Z}^{k},~ Σ_{i=0}^{k-1}|λ_{i}|=h \},\] 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.

14 pages