Direct and Inverse Problems for Restricted Signed Sumsets -- I

Let $A=\{a_{1},\ldots,a_{k}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$, the $h$-fold signed sumset of $A$, denoted by $h_{\pm}A$, is defined as $$h_{\pm}A=\left\lbrace \sum_{i=1}^{k} \lambda_{i} a_{i}: \lambda_{i} \in \{-h, \ldots, 0, \ldots, h\} \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left|\lambda_{i} \right| =h\right\rbrace,$$ and the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A=\left\lbrace \sum_{i=1}^{k} \lambda_{i} a_{i}: \lambda_{i} \in \left\lbrace -1, 0, 1\right\rbrace \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left|\lambda_{i} \right| = h\right\rbrace.$$ A direct problem for the sumset $h^{\wedge}_{\pm}A$ is to find the optimal size of $h^{\wedge}_{\pm}A$ in terms of $h$ and $|A|$. An inverse problem for this sumset is to determine the structure of the underlying set $A$ when the sumset $h^{\wedge}_{\pm}A$ has optimal size. While some results are known for the signed sumsets in finite abelian groups due to Bajnok and Matzke, not much is known for the restricted $h$-fold signed sumset $h^{\wedge}_{\pm}A$ even in the additive group of integers $\Bbb Z$. In case of $G = \Bbb Z$, Bhanja, Komatsu and Pandey studied these problems for the sumset $h^{\wedge}_{\pm}A$ for $h=2, 3$, and $k$, and conjectured the direct and inverse results for $h \geq 4$. In this paper, we prove these conjectures completely for the sets of positive integers. In a subsequent paper, we prove these conjectures for the sets of nonnegative integers.