Nested Cantor sets

TL;DR

This paper establishes conditions under which two Cantor sets are nested for many translations, with applications in diophantine approximation and hyperbolic attractors, and proves the optimality of these conditions for natural sets.

Contribution

It provides sufficient and optimal conditions for nesting of Cantor sets under translation, connecting diophantine approximation and dynamical systems.

Findings

Conditions for nesting are established and shown to be optimal for natural Cantor sets.

The problem relates to diophantine approximation and hyperbolic attractors.

The results offer a toy model for parameter selection in complex dynamical systems.

Abstract

We give sufficient conditions for two Cantor sets of the line to be nested for a positive set of translation parameters. This problem occurs in diophantine approximations. It also occurs as a toy model of the parameter selection for non-uniformly hyperbolic attractors of the plane. For natural Cantors sets, we show that this condition is optimal.