A non-injective Assouad-type theorem with sharp dimension

Lipschitz light maps, defined by Cheeger and Kleiner, are a class of non-injective “foldings” between metric spaces that preserve some geometric information. We prove that if a metric space $(X,d)$ has Nagata dimension $n$, then its “snowflakes” $(X,d^{\epsilon })$ admit Lipschitz light maps to ${\mathbb{R}}^n$ for all $0<\epsilon <1$. This can be seen as an analog of a well-known theorem of Assouad. We also provide an application to a new variant of conformal dimension.