Some direct and inverse problems for the Restricted Signed sumset in set of integers

Given a positive integer h and a nonempty finite set of integers A=₁,a₂,…,aₖ\, the restricted h-fold signed sumset of A, denoted by h^±A, is defined as $$h^{}±A= ∑ᵢ₌₁ᵏ λᵢ aᵢ: λᵢ ∈ -1, 0, 1 \ for \ i= 1, 2, …, k \ and \ ∑ᵢ₌₁ᵏ | λᵢ | =h.$$ The direct problem associated with this sumset is to find the optimal lower bound of $|h^{}±A|$, and the inverse problem associated with this sumset is to determine the structure of the underlying set $A$, when $|h^{}±A|$ attains the optimal lower bound. Bhanja, Komatsu and Pandey studied the direct and inverse problem for the restricted $h$-fold signed sumset for $h=2, 3$, and $k$ and conjectured some direct and inverse results for $h ≥ 4$. In this paper, we prove these conjectures for $h=4$. We also prove the direct and inverse theorems for arbitrary $h$ under certain restrictions on the set $A$ which are particular cases of the conjectures. Moreover, we prove these conjectures for arithmetic progressions.