Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets

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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 Rᵈ. 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≥ 1. This partially answers a question raised by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223--233].