Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets.

The higher dimensional Frobenius problem was introduced by a preceding paper [Fan et al. (2015) [8] ]. In this paper, we investigate the Lipschitz equivalence of dust-like self-similar sets in 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 ≥ 1 . This partially answers a question raised by Falconer and Marsh (1992) [7] . [ABSTRACT FROM AUTHOR]