Higher dimensional Frobenius problem: Maximal saturated cone, growth function and rigidity

2014/11/26 by Ai-hua Fan, Aihua Fan, Fan, Ai-hua +4 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems #math.NT #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1411.7118

arxiv created 2014/11/26 · openalex publication_date 2014/11/26 · arxiv updated 2014/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider m integral vectors X₁,...,X_m ∈ ℤ^s located in a half-space of ℝ^s (m≥ s≥ 1) and study the structure of the additive semi-group X₁ ℕ +... + X_m ℕ. We introduce and study maximal saturated cone and directional growth function which describe some aspects of the structure of the semi-group. When the vectors X₁, ..., X_m are located in a fixed hyperplane, we obtain an explicit formula for the directional growth function and we show that this function completely characterizes the defining data (X₁, ..., X_m) of the semi-group. The last result will be applied to the study of Lipschitz equivalence of Cantor sets (see [H. Rao and Y. Zhang, Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets, Preprint 2014]).

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• Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets

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