Is it true that if $X$ has packing dimension $\alpha$, then we can write $X$ as the countable union of sets $X_i$, where $X_i$ has Minkowski dimension $\alpha$. If not, which notion of dimension is closest to a countably stable version of Minkowski dimension, and why?
2026-03-25 09:27:03.1774430823
Packing Dimension as a Countable Union of Minkowski Dimension Sets
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I think I've discovered an answer to the question. The packing dimension is equivalent to the upper modified box counting dimension, which is defined for a set $E$ as
$$ \inf \{ \sup( \dim_M(E_i)): E \subset E_i \} $$
This shows that every set with a given packing dimension can be written as a countable union of sets with that Minkowski dimension.