We show if $K$ is a self-similar $1$-set that either satisfies the strong separation 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.
Local Geometry of Self-similar Sets: Typical Balls, Tangent Measures and Asymptotic Spectra
Dimension of ergodic measures projected onto self-similar sets with overlaps
New results on embeddings of self-similar sets via renormalization
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