Gap sequence, Lipschitz equivalence and box dimension of fractal sets

Abstract

We introduce a notion of gap sequences for compact sets , which is a generalization of the gap sequences of compact sets on the real line. We show that if the gap sequences of two fractal sets are not equivalent, then these two sets cannot be Lipschitz equivalent, where the latter fact is usually very hard to verify. Finally, we show that for some typical fractal sets, the gap sequences characterize the upper box dimension.