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

2024/03/06 by Mohan, Mistri, Raj Kumar, Pandey, Ram Krishna
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2403.03625

Abstract

Given a positive integer h and a nonempty finite set of integers A=\a₁,a₂,…,a_k\, the restricted h-fold signed sumset of A, denoted by h^\wedge_±A, is defined as h^\wedge_±A=\lbrace ∑_i=1^k λ_i a_i: λ_i ∈ \lbrace -1, 0, 1\rbrace for i= 1, 2, …, k and ∑_i=1^k | λ_i | =h\rbrace. The direct problem associated with this sumset is to find the optimal lower bound of |h^\wedge_±A|, and the inverse problem associated with this sumset is to determine the structure of the underlying set A, when |h^\wedge_±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.

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