Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets

Lipschitz Equivalence of Cantor Sets and Algebraic Properties of Contraction Ratios

In this paper we investigate the Lipschitz equivalence of dust-like self-similar sets in Rd. One of the fundamental results by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223– 233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. In this paper we extend the study by...

Cantor Sets, Binary Trees and Lipschitz Circle Homeomorphisms

We define the notion of rotations on infinite binary trees, and construct an irrational tree rotation with bounded distortion. This lifts naturally to a Lipschitz circle homeomorphism having the middle-thirds Cantor set as its minimal set. This degree of smoothness is best possible, since it is known that no C1 circle diffeomorphism can have a linearly self-similar Cantor set as its minimal set.

Uncountably Many Inequivalent Lipschitz Homogeneous Cantor Sets in Ir

The Multi-Dimensional Frobenius Problem

Consider the problem of determining maximal vectors g such that the Diophantine system Mx = g has no solution. We provide a variety of results to this end: conditions for the existence of g, conditions for the uniqueness of g, bounds on g, determining g explicitly in several important special cases, constructions for g, and a reduction for M .

Cantor sets

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ω of irrationals, or certain of its subspaces. In particular, given f ∈ (ω\{0}), we consider compact sets of the form Q i∈ω Bi, where |Bi| = f(i) for all, or for infinitely many, i. We also consider “n-splitting” compact sets, i.e., compact sets K such that for any f ∈ K and i ∈ ω, |{g(i) : ...