# Robustness of doubling dimension to small perturbations

Let $$M$$ be a metric space. Then the doubling dimension of $$M$$, denoted $$\dim M$$, is defined to be the minimum value $$k$$ such that every ball in $$M$$ of radius $$r$$ can be covered by at most $$2^k$$ balls of radius $$r/2$$.

In the paper "Small Hop-diameter Sparse Spanners for Doubling Metrics" by Chan/Gupta, the authors write:

Apart from being a generalization of the $$\ell_p$$ notion of dimension, designing algorithms that only use the doubling properties (instead of the geometry of $$\mathbb R^k$$) has other advantages: the notion of doubling dimension is fairly resistant to small perturbations in the distances: for instance, if one takes a distance matrix of a set of points in $$\ell_p^k$$ and slightly changes some of the entries, then the doubling dimension does not change by much, but the metric may not remain isometrically embeddable in $$\ell_p$$ (into any number of dimensions).

1. Is there a known formalization of this stability, perhaps with respect to the Hausdorff distance on (finite) subsets of $$\ell_p^k$$? That is, do we have something like the following: Let $$A$$ and $$B$$ be finite subsets of $$\ell_p^k$$. Then $$\lvert\dim A - \dim B\rvert \leq d_\text H(A,B)$$, where $$d_\text H$$ is the Hausdorff distance?

2. Does this stability generalize to arbitrary finite metric spaces, with respect to, e.g., the Gromov–Hausdorff distance? That is, if $$A$$ and $$B$$ are finite metric spaces, do we have something like $$\lvert\dim A - \dim B\rvert \leq d_\text{GH}(A,B)$$, where $$d_\text{GH}$$ is the Gromov–Hausdorff distance?

If 'No' to the above, are there any sensible formalizations of Chan & Gupta's statement that make their claim of stability precise?

I have come to the conclusion that my questions $$1$$ and $$2$$ can be seen to be false by considering simple examples in $$\mathbb R$$. Morally, since doubling dimension takes only integer values, a general stability result is not possible in the sense that I describe.
However, after further reading, I believe what Chan & Gupta are referring to is the robustness result given as Propsition $$3$$ in "Bypassing the Embedding" by Talwar:
Proposition 3 [Robustness]. Let $$(X, d)$$ and $$(Y, d')$$ be metric spaces such that there is a bijective map $$f: X \rightarrow Y$$ satisfying $$d\left(x_1, x_2\right) \leq$$ $$d' \left(f\left(x_1\right), f\left(x_2\right)\right) \leq D \cdot d\left(x_1, x_2\right)$$. Then $$k_Y \leq 2 k_X \cdot\lceil\log 4 D\rceil$$.
Note that Talwar denotes the doubling dimension of $$X$$ by $$k_{X}$$.