Interval projections of self-similar sets

TL;DR

This paper proves that for certain self-similar 1-sets in the plane, only finitely many lines through the origin project the set onto an interval, under specific separation or homothety conditions.

Contribution

It establishes a finiteness result for interval projections of self-similar sets under particular geometric and separation conditions.

Findings

Finitely many lines through the origin produce interval projections of the set.

The result applies to self-similar sets not contained in a line.

Conditions include strong separation or definition via homotheties.

Abstract

We show that if $K$ is a self-similar $1$-set that is not contained in a line and either satisfies the strong separation condition or is defined via homotheties then there are at most finitely many lines through the origin such that the projection of $K$ onto them is an interval.