While reading around online earlier today, I managed to stumble onto a strange claim, that I have not been able to find or produce a proof for. The claim is that if $X$ is a compact subset of $\mathbb R^n$, which can be written as a disjoint union of some $X_i$ where $X_i$ is isometric to a dilation of $X$ by some constant $c_i$, then the Hausdorff dimension has the property that $\sum c_i^d = 1$.
I know the definition of Hausdorff dimension, and so I was able to verify that this is true for some examples that I know off the top of my head, i.e. Cantor sets in $\mathbb R$, Sierpinski fractals, etc. But I have not been able to come up with a proof.
I expect it is not a hard theorem, because a google search tells me that more technical versions of this theorem can be stated, with all kinds of overlap conditions on the $X_i$ and whatnot, but none of this is necessary, so somehow I think it should not be hard.. but I cannot figure it out.
Any help is appreciated.