Sums of two self-similar Cantor sets

TL;DR

This paper proves that for self-similar Cantor sets with combined Hausdorff dimension exceeding 1, small perturbations can generate an interval in their sumset, supporting a weaker form of the Palis conjecture.

Contribution

It demonstrates that small perturbations of self-similar Cantor sets can produce intervals in their sumset when the sum of their Hausdorff dimensions exceeds 1.

Findings

Sumset contains an interval after perturbation

Supports a weaker form of the Palis conjecture

Perturbations have more freedom than in previous settings

Abstract

We show that for any pair of self-similar Cantor sets with sum of Hausdorff dimensions greater than 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of self-similar Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis conjecture.