Direct and Inverse Problems for Restricted Signed Sumsets -- I

2025/04/12 by Raj Kumar Mistri, Mistri, Raj Kumar, Nitesh Prajapati +1
Computer Science · Mathematics · #Matrix Theory and Algorithms #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2504.09316

Abstract

Let A=\a₁,…,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_±A, is defined as h_±A=\lbrace ∑_i=1^k λ_i a_i: λ_i ∈ \-h, …, 0, …, h\ for i= 1, 2, …, k and ∑_i=1^k |λ_i | =h\rbrace, and 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. A direct problem for the sumset h^\wedge_±A is to find the optimal size of h^\wedge_±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_±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_±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_±A for h=2, 3, and k, and conjectured the direct and inverse results for h ≥ 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.

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