  We study the problem of embedding minimal dynamical systems into the shift
action on the Hilbert cube $\left([0,1]^N\right)^{\mathbb{Z}}$. This problem is
intimately related to the theory of mean dimension, which counts the averaged
number of parameters of dynamical systems. Lindenstrauss proved that minimal
systems of mean dimension less than $N/36$ can be embedded into
$\left([0,1]^N\right)^{\mathbb{Z}}$, and he proposed the problem of finding the
optimal value of the mean dimension for the embedding. We solve this problem by
proving that minimal systems of mean dimension less than $N/2$ can be embedded
into $\left([0,1]^N\right)^{\mathbb{Z}}$. The value $N/2$ is optimal. The proof
uses Fourier and complex analysis.
