  We prove that every $\mathbb{Z}^{k}$-action $(X,\mathbb{Z}^{k},T)$ of mean
dimension less than $D/2$ admitting a factor $(Y,\mathbb{Z}^{k},S)$ of Rokhlin
dimension not greater than $L$ embeds in
$(([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}\times Y,\sigma\times S)$, where
$D\in\mathbb{N}$, $L\in\mathbb{N}\cup\{0\}$ and $\sigma$ is the shift on the
Hilbert cube $([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}$; in particular, when
$(Y,\mathbb{Z}^{k},S)$ is an irrational $\mathbb{Z}^{k}$-rotation on the
$k$-torus, $(X,\mathbb{Z}^{k},T)$ embeds in
$(([0,1]^{2^kD+1})^{\mathbb{Z}^k},\sigma)$, which is compared to a previous
result by the first named author, Lindenstrauss and Tsukamoto. Moreover, we
give a complete and detailed proof of Takens' embedding theorem with a
continuous observable for $\mathbb{Z}$-actions and deduce the analogous result
for $\mathbb{Z}^{k}$-actions. Lastly, we show that the Lindenstrauss--Tsukamoto
conjecture for $\mathbb{Z}$-actions holds generically, discuss an analogous
conjecture for $\mathbb{Z}^{k}$-actions appearing in a forthcoming paper by the
first two authors and Tsukamoto and verify it for $\mathbb{Z}^{k}$-actions on
finite dimensional spaces.
