A realization result for systems of sets of lengths

2020/08/20 by Alfred Geroldinger, Geroldinger, Alfred, Qinghai Zhong +1
Mathematics · #13A05 #13F05 #20M13 #Algebraic Geometry and Number Theory #Combinatorics #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #Dedekind cut #Discrete mathematics #Domain (mathematical analysis) #FOS: Mathematics #Mathematical analysis #Mathematics #Physics #Realization (probability) #Rings, Modules, and Algebras #math.AC #msc:13A05 #msc:13F05 #msc:20M13

paper · pdf · doi:10.48550/arxiv.2008.08820

To appear in Israel Journal of Mathematics

openalex publication_date 2020/08/20 · arxiv created 2021/01/15 · arxiv updated 2021/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathcal L^* be a family of finite subsets of \mathbb N₀ having the following properties. (a). \0\, \1\ ∈ \mathcal L^* and all other sets of \mathcal L^* lie in \mathbb N_{≥ 2}. (b). If L₁, L₂ ∈ \mathcal L^*, then the sumset L₁ + L₂ ∈ \mathcal L^*. We show that there is a Dedekind domain D whose system of sets of lengths equals \mathcal L^*.

Citations

• Factorization in Krull monoids with infinite class group
• Every abelian group is a class group
• Factorization theory in commutative monoids

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• Long arithmetic progressions in sumsets: Thresholds and Bounds
• Sets of Lengths
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• The Typical Structure of Sets with Small Sumset