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Higher dimensional Frobenius problem: maximal saturated cone, growth function and rigidity. (English. French summary) Zbl 06473577
Summary: We consider $m$ integral vectors $X_1, \ldots, X_m \in \mathbb{Z}^s$ located in a half-space of $\mathbb{R}^s$ ($m \geq s \geq 1$) and study the structure of the additive semi-group $X_1 \mathbb{N} + \cdots + X_m \mathbb{N}$. 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 $X_1, \cdots, X_m$ 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 $(X_1, \cdots, X_m)$ of the semi-group. The last result will be applied to the study of Lipschitz equivalence of Cantor sets (see [11]).
MSC:
11B05Topology etc. of sets of numbers
52C99Discrete geometry
37A35Entropy and other invariants, isomorphism, classification (ergodic theory)
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