Direct and Inverse Theorems on Signed Sumsets of Integers

2018/10/05 by Bhanja, Jagannath, Pandey, Ram Krishna
#11B75 #11P70 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1810.02673

Abstract

Let G be an additive abelian group and h be a positive integer. For a nonempty finite subset A=\a₀, a₁,…, a_k-1\ of G, we let h_\underline+A:=\Σ_i=0^k-1λ_i a_i: (λ₀, …, λ_k-1) ∈ ℤ^k,~ Σ_i=0^k-1|λ_i|=h \, be the \it signed sumset of A. The \it direct problem for the signed sumset h_\underline+A is to find a nontrivial lower bound for |h_\underline+A| in terms of |A|. The \it 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 both the direct and inverse problems for |h_\underline+A|, when A is a finite set of integers.

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