In the seminal paper Mean dimension, small entropy
factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math
89 (1999) 227-262, E. Lindenstrauss initiated the study of the
relation between embeddability and mean dimension and raised the
corresponding question for (minimal) $Z^k$-actions. We solve the problem
by showing that minimal systems of mean dimension less than $N/2$
embed equivariantly into the cubical shift $([0,1]^N)^{Z^k}$. The result
is optimal and improves upon a previous result by Gutman,
Lindenstrauss and Tsukamoto. The proof uses multi-dimensional sampling
theory and is significantly harder than the proof of the case $k=1$
which was settled previously by Gutman and Tsukamoto. Joint work with
Yixiao Qiao and Masaki Tsukamoto.