# Equi-Hölder embeddings of compact metric spaces of finite packing dimension into $\ell_2$

Problem. Does a compact metric space of finite packing dimension admit an equi-Hölder embedding into a Hilbert space?

A map $f:X\to Y$ between metric spaces $(X,d_X)$, $(Y,d_Y)$ is called equi-Hölder embedding if there are positive real constants $c,C,\alpha$ such that $$c\cdot d_X(x,y)^\alpha\le d_Y(f(x),f(y))\le C\cdot d_X(x,y)^\alpha$$ for all $x,y\in X$.

The packing dimension of a compact metric space $(X,d)$ in the (finite or infinite) number $$Dim(X)=\limsup_{\varepsilon\to 0}\frac{\ln N_\varepsilon(X)}{\ln(1/\varepsilon)},$$ where $N_\varepsilon(X)$ is the cardinality of the smallest cover of $X$ by subsets of diameter $\le\varepsilon$.

Remark. By the Assouad Embedding Theorem, a metric space $X$ admits an equi-Hölder embedding in a finite-dimensional Hilbert space if and only if $X$ is doubling. It can be shown that doubling metric spaces have finite packing dimension.

If $X$ has packing dimension (in your sense; they call it "fractal dimension") less than $m/2$, then $X$ has an embedding $f$ into $\mathbb{R}^m$ such that $$L d(x,y)^\gamma \leq |f(x)-f(y)| \leq d(x,y)$$ for all $x,y\in X$, where $L>0$ and $\gamma\geq 1$.

This is not the same as what you ask, but it is the only result of this type that I know that assumes finite packing dimension rather than finite Assouad dimension.

[1] C. Foias, E. Olson, E. Finite fractal dimension and Hölder-Lipschitz parametrization. Indiana Univ. Math. J. 45 (1996), no. 3, 603–616.

[2] J. Robinson, Dimensions, embeddings, and attractors. Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.