Direct and inverse problems for restricted signed sumsets in integers

Abstract

Let $A=\{a_0, a_1,\ldots, a_{k-1}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$ $(\leq k)$, we let

$h^{\wedge}_{\pm}A = \{\Sigma_{i=0}^{k-1} \lambda_{i} a_{i}: \lambda_{i} \in \{-1,0,1\} \text{ for } i=0, 1, \ldots, k-1,~~\Sigma_{i=0}^{k-1} |\lambda_{i}|=h\},$

be the $h$-fold restricted signed sumset of $A$. The direct problem for the restricted signed sumset is to find the minimum number of elements in $h^{\wedge}_{\pm}A$ in terms of $\lvert A\rvert$, where $\lvert A\rvert$ is the cardinality of $A$. The {\it inverse problem} for the restricted signed sumset is to determine the structure of the finite set $A$ for which the minimum value of $|h^{\wedge}_{\pm}A|$ is achieved. In this article, we solve some cases of both direct and inverse problems for $h^{\wedge}_{\pm}A$ in the group of integers. In this connection, we also mention some conjectures in the remaining cases.