Take the tour ×  Let $G$ be a metric group, and let $h$ be the
associated Hausdorff dimension function on subsets of $G$. (See
for instance Barnea and Shalev, Hausdorff dimension, pro-p
groups and Kac-Moody algebras, Trans. AMS 1997.) When do we have
$h(AB) = h(A) + h(B) - h(A \cap B)$ for normal subgroups of $G$?
If this property fails, is there still a general way to use $h$
(or some similar 'dimension' function) to construct a
pseudometric on the lattice of normal subgroups of $G$? What I
am looking for here are some informative examples of bad
behaviour, particularly for profinite groups.  share|improve
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accepted  In the first conference I ever went to Slava
Grigorchuk asked me a similar question and I didn’t have an
answer. But when I have got back to Jerusalem I have talked with
Elon Lindenstrauss about it and he suggested the following easy
counterexample. Take $G=\mathbb{F}_p[[t]]$. Pick $S$ to be a
subset of the integers with density one and wit