Open intervals in sums and products of Cantor sets

TL;DR

This paper introduces new methods to demonstrate that sums and products of multiple central Cantor sets can generate large open intervals, extending to their $C^1$ images and related power sets.

Contribution

It provides novel arguments for the formation of open intervals from sums and products of Cantor sets, offering a different perspective from previous thickness-based approaches.

Findings

Sums and products of sufficient Cantor sets produce large open intervals.

Results extend to $C^1$ images of Cantor sets.

Addresses recent questions on powers of Cantor sets.

Abstract

We give new arguments for sums and products of sufficient numbers of arbitrary central Cantor sets to produce large open intervals. We further discuss the same question for $C^1$ images of such central Cantor sets. This gives another perspective on the results obtained by Astels through a different formulation on the thickness of these Cantor sets. There has been recent interest in the question of products and sums of powers of Cantor sets, and these are addressed by our methods.