Direct and inverse results on restricted signed sumsets in integers

Abstract: Let $G$ be an additive abelian group. Let $A={a_{0}, a_{1},\ldots, a_{k-1}}$ be a nonempty finite subset of $G$. For a positive integer $h$ satisfying $1\leq h\leq k$, we let [h\hat{}{\underline{+}}A:={\Sigma{i=0}^{k-1}\lambda_{i} a_{i}: (\lambda_{0},\lambda_{1}, \ldots, \lambda_{k-1}) \in {-1,0,1}^{k},~\Sigma_{i=0}^{k-1}|\lambda_{i}|=h },] be the restricted signed sumset of $A$. The direct problem for the restricted signed sumset $h\hat{}{\underline{+}}A$ is to find the minimum number of elements in $h\hat{}{\underline{+}}A$ in terms of $|A|$. The inverse problem for $h\hat{}{\underline{+}}A$ is to determine the structure of the finite set $A$ for which $|h\hat{}{\underline{+}}A|$ is minimal. In this article, we solve some cases of both direct and inverse problems for $h\hat{}_{\underline{+}}A$, when $A$ is a finite set of integers. In this connection, we also pose some questions as conjectures in the remaining cases.