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

TL;DR

This paper investigates the structure of additive semi-groups generated by integral vectors in higher dimensions, introducing concepts like maximal saturated cones and growth functions, with explicit formulas and applications to Lipschitz equivalence of Cantor sets.

Contribution

It introduces the maximal saturated cone and directional growth function for higher-dimensional Frobenius problems, providing explicit formulas and characterizations.

Findings

Explicit formula for the directional growth function when vectors lie in a hyperplane

The growth function characterizes the semi-group's defining data

Applications to Lipschitz equivalence of Cantor sets

Abstract

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