Direct and inverse results on restricted signed sumsets in integers

Abstract

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

18 pages