On the interior of projections of planar self-similar sets

TL;DR

This paper demonstrates that projections of planar self-similar sets with Hausdorff dimension greater than 1 can be perturbed slightly to produce sets with nonempty interior, under the open set condition.

Contribution

It shows how small perturbations can induce interior in projections of self-similar sets satisfying specific conditions, advancing understanding of their geometric properties.

Findings

Projections can have nonempty interior after small perturbations.

Open set condition is crucial for creating interior in projections.

Hausdorff dimension > 1 is a key requirement.

Abstract

We consider projections of planar self-similar sets, and show that one can create nonempty interior in the projections by applying arbitrary small perturbations, if the self-similar set satisfies the open set condition and has Hausdorff dimension greater than 1.