  Realization of $d$-dimensional effective subshifts as projective sub-actions
of $d+d'$-dimensional sofic subshifts for $d'\geq 1$ is now well
know~\cite{Hochman-2009,Durand-Romashchenko-Shen-2010,Aubrun-Sablik-2010}. In
this paper we are interested in the speed of convergence of this realization.
That is to say given an effective subshift $\Sigma$ realized as projective
sub-action of a sofic $\T$, we study the function which on input an integer $k$
returns the smallest width of the strip which verify the local rules of $\T$
necessary to obtain exclusively the language of size $k$ of $\Sigma$ in the
central row of the strip. We study this topological conjugacy invariant for
effective subshifts in order to exhibit algorithmic properties of these
subshifts.
