  Mean dimension is a topological invariant of dynamical systems, which
originates with Mikhail Gromov in 1999 and which was studied with deep
applications around 2000 by Elon Lindenstrauss and Benjamin Weiss within the
framework of amenable group actions. Let a countable discrete amenable group
$G$ act continuously on compact metrizable spaces $X$ and $Y$. Consider the
product action of $G$ on the product space $X\times Y$. The product inequality
for mean dimension is well known: $\mathrm{mdim}(X\times
Y,G)\le\mathrm{mdim}(X,G)+\mathrm{mdim}(Y,G)$, while it was unknown for a long
time if the product inequality could be an equality. In 2019, Masaki Tsukamoto
constructed the first example of two different continuous actions of $G$ on
compact metrizable spaces $X$ and $Y$, respectively, such that the product
inequality becomes strict. However, there is still one longstanding problem
which remains open in this direction, asking if there exists a continuous
action of $G$ on some compact metrizable space $X$ such that
$\mathrm{mdim}(X\times X,G)<2\cdot\mathrm{mdim}(X,G)$. We solve this problem.
Somewhat surprisingly, we prove, in contrast to (topological) dimension theory,
a rather satisfactory theorem: If an infinite (countable discrete) amenable
group $G$ acts continuously on a compact metrizable space $X$, then we have
$\mathrm{mdim}(X^n,G)=n\cdot\mathrm{mdim}(X,G)$, for any positive integer $n$.
Our product formula for mean dimension, together with the example and
inequality (stated previously), eventually allows mean dimension of product
actions to be fully understood.
