Embedding minimal dynamical systems into Hilbert cubes

We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube ([0,1]N)Z. This problem is intimately related to the theory of mean dimension, which counts the average number of parameters for describing a dynamical system. Lindenstrauss proved that minimal systems of mean dimension less than cN for c= 1 / 36 can be embedded in ([0,1]N)Z, and asked what is the optimal value for c. We solve this problem by showing embedding is possible when c= 1 / 2. The value c= 1 / 2 is optimal. The proof exhibits a new interaction between harmonic analysis and dynamical coding techniques.