  We estimate the growth of matter perturbations in a class of recently
proposed dark-energy models based on the (loop-corrected) gravi-dilaton string
effective action, and characterized by a global attractor epoch in which
dark-matter and dark-energy density scale with the same effective equation of
state. Unlike most dark-energy models, we find that the accelerated phase might
start even at redshifts as high as z~5 (thus relaxing the coincidence problem),
while still producing at present an acceptable level of matter fluctuations. We
also show that such an early acceleration is not in conflict with the recently
discovered supernova SN1997ff at z~1.7 . The comparison of the predicted value
of sigma_{8} with the observational data provides interesting constraints on
the fundamental parameters of the given model of dilaton-dark matter
interactions.
