From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets

Let $K,F\subset\mathbb{R}^d$ be two dust-like self-similar sets sharing the same Hausdorff dimension. We consider when the mere existence of a Lipschitz embedding from $K$ to $F$ already implies their Lipschitz equivalence. Our main result is threefold: (1) if the Lipschitz image of $K$ intersects $F$ in a set of positive Hausdorff measure, then $K$ admits a Lipschitz surjection onto $F$; (2) if $F$ is in addition homogeneous, then the generating iterated function systems of $K, F$ should have algebraically dependent ratios and consequently, $K$ and $F$ are Lipschitz equivalent; (3) the Lipschitz equivalence can fail without the homogeneity assumption. This answers two questions in Balka and Keleti [Adv. Math. 446 (2024), 109669].