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" $(X,d^ε)$ admit Lipschitz light maps to $\mathbb{R}^n$ for all $0<ε<1$. 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.

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