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=\a₁, a₂, , a₊\, the restricted h-fold signed sumset of A, denoted by h^_A, is defined as h^_A= ₈=₁^k ₈ a₈: ₈ -1, 0, 1 \ for \ i= 1, 2, , k \ and \ ₈=₁^k | ₈ | =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.