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

Abstract

We consider m integral vectors X1,…,Xm∈Zs located in a half-space of Rs (m≥s≥1) and study the structure of the additive semi-group X1N+⋯+XmN. 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,⋯,Xm 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,⋯,Xm) of the semi-group. The last result will be applied to the study of Lipschitz equivalence of Cantor sets (see [11]).