Sums of two homogeneous Cantor sets

TL;DR

This paper proves that for certain homogeneous Cantor sets with combined Hausdorff dimension exceeding 1, small perturbations can produce intervals in their sumset, providing evidence towards a weaker version of Palis' conjecture.

Contribution

It demonstrates that small perturbations can generate intervals in the sumset of homogeneous Cantor sets with dimension sum over 1, advancing understanding of sumset structures.

Findings

Small perturbations create intervals in sumsets of homogeneous Cantor sets with dimension sum > 1.

The result applies to self-similar sets with overlaps and $L^2$-density measures.

Provides an affirmative answer to a weaker form of Palis' conjecture.

Abstract

We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous 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. We also consider self-similar sets with overlaps on the real line (not necessarily homogeneous), and show that one can create an interval by applying arbitrary small perturbations, if the uniform self-similar measure has $L^2$-density.