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

We consider $m$ integral vectors $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m\in {\mathbb{Z}}^s$ located in a half-space of ${\mathbb{R}}^s$ ($m\ge s\ge 1$) and study the structure of the additive semi-group $X_1\mathbb{N}+\mathrm{.}\mathrm{.}\mathrm{.}+X_m\mathbb{N}$. 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 $X_1, …, X_m$ 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 $(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m)$ 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]).