  We analyze the transmitted flux in a sample of 17 QSOs spectra at
5.74<zem<6.42 to obtain tighter constraints on the volume-averaged neutral
hydrogen fraction, xHI, at z~6. We study separately the narrow transmission
windows (peaks) and the wide dark portions (gaps) in the observed absorption
spectra. By comparing the statistics of these spectral features with Lyalpha
forest simulations, we conclude that xHI evolves smoothly from 10^{-4.4} at
z=5.3 to 10^{-4.2} at z=5.6, with a robust upper limit xHI<0.36 at z=6.3. The
frequency and physical sizes of the peaks imply an origin in cosmic underdense
regions and/or in HII regions around faint quasars or galaxies. In one case
(the intervening HII region of the faint quasar RD J1148+5253 at z=5.70 along
the LOS of SDSS J1148+5251 at z=6.42) the increase of the peak spectral density
is explained by the first-ever detected transverse proximity effect in the HI
Lyalpha forest; this indicates that at least some peaks result from a locally
enhanced radiation field. We then obtain a strong lower limit on the foreground
QSO lifetime of tQ>11 Myr. The observed widths of the peaks are found to be
systematically larger than the simulated ones. Reasons for such discrepancy
might reside either in the photoionization equilibrium assumption or in
radiative transfer effects.
