Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets

TL;DR

This paper links the higher dimensional Frobenius problem to the Lipschitz equivalence of certain self-similar fractal sets, providing new invariants and criteria for their classification.

Contribution

It introduces the directional growth function of the Frobenius problem as a Lipschitz invariant and characterizes Lipschitz equivalence of dust-like self-similar sets with coplanar ratios.

Findings

Directional growth function is a Lipschitz invariant.

Lipschitz equivalence characterized by permutation of contraction vectors.

Partially answers a longstanding question on Cantor set equivalence.

Abstract

The higher dimensional Frobenius problem was introduced by a preceding paper [Fan, Rao and Zhang, Higher dimensional Frobenius problem: maximal saturated cones, growth function and rigidity, Preprint 2014]. %the higher dimensional Frobenius problem was introduced and a directional growth function was studied. In this paper, we investigate the Lipschitz equivalence of dust-like self-similar sets in $\mathbb R^d$. For any self-similar set, we associate with it a higher dimensional Frobenius problem, and we show that the directional growth function of the associate higher dimensional Frobenius problem is a Lipschitz invariant. As an application, we solve the Lipschitz equivalence problem when two dust-like self-similar sets $E$ and $F$ have coplanar ratios, by showing that they are Lipschitz equivalent if and only if the contraction vector of the $p$-th iteration of $E$ is a permutation of that of the $q$-th iteration of $F$ for some $p, q\geq 1$. This partially answers a question raised by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, \emph{Mathematika,} \textbf{39} (1992), 223--233].