Products of two Cantor sets

TL;DR

This paper investigates conditions under which the product of two Cantor sets forms an interval, with implications for understanding spectra in quasicrystal models and intersections of Cantor sets.

Contribution

It provides optimal thickness estimates ensuring the product of two Cantor sets is an interval, advancing the understanding of their structural properties.

Findings

Derived optimal thickness bounds for product intervals

Connected Cantor set products to quasicrystal spectra

Discussed intersections of Cantor sets in relation to the problem

Abstract

We consider products of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by a product of two Cantor sets. We also discuss the connection with the question on the structure of intersections of two Cantor sets, which was considered by many authors previously.