  We present results from a new Keck spectroscopic survey of UV-faint LBGs in
the redshift range 3<z<7. Combined with earlier Keck and published ESO VLT
data, our sample contains more than 600 dropouts, offering new insight into the
nature of sub-L* sources typical of those likely to dominate the cosmic
reionisation process. Here we use this sample to characterise the fraction of
strong Lya emitters within the continuum-selected dropouts. By quantifying how
the "Lya fraction" varies with redshift, we seek to constrain changes in Lya
transmission associated with reionisation. In order to distinguish the effects
of reionisation from other factors which affect the Lya fraction (e.g. dust,
ISM kinematics), we study the luminosity and redshift-dependence of the Lya
fraction over 3<z<6, when the IGM is known to be ionised. These results reveal
that low luminosity galaxies show strong Lya emission much more frequently than
luminous systems, and that at fixed luminosity, the prevalence of strong Lya
emission increases moderately with redshift over 3 < z < 6. Based on the
correlation between blue UV slopes and strong Lya emitting galaxies in our
dataset, we argue that the Lya fraction trends are governed by redshift and
luminosity-dependent variations in the dust obscuration, with likely additional
contributions from trends in the kinematics and covering fraction of neutral
hydrogen. We find a tentative decrease in the Lya fraction at z~7 based on the
limited IR spectra for candidate z~7 lensed LBGs, a result which, if confirmed
with future surveys, would suggest an increase in the neutral fraction by this
epoch. Given the supply of z and Y-drops now available from Hubble WFC3/IR
surveys, we show it will soon be possible to significantly improve estimates of
the Lya fraction using optical and near-IR spectrographs, thereby extending the
study conducted in this paper to 7<z<8.
