Direct and inverse results on restricted signed sumsets in integers

2019/07/31 by Bhanja, Jagannath, Komatsu, Takao, Pandey, Ram Krishna
#11B13 #11B75 #11P70 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1908.00081

Abstract

Let G be an additive abelian group. Let A=\a₀, a₁,…, a_k-1\ be a nonempty finite subset of G. For a positive integer h satisfying 1≤ h≤ k, we let h_\underline+A:=\Σ_i=0^k-1λ_i a_i: (λ₀,λ₁, …, λ_k-1) ∈ \-1,0,1\^k,~Σ_i=0^k-1|λ_i|=h \, be the restricted signed sumset of A. The direct problem for the restricted signed sumset h_\underline+A is to find the minimum number of elements in h_\underline+A in terms of |A|. The inverse problem for h_\underline+A is to determine the structure of the finite set A for which |h_\underline+A| is minimal. In this article, we solve some cases of both direct and inverse problems for h_\underline+A, when A is a finite set of integers. In this connection, we also pose some questions as conjectures in the remaining cases.

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