  The reionization of the Universe is one of the most important topics of
present day astrophysical research. The most plausible candidates for the
reionization process are star-forming galaxies, which according to the
predictions of the majority of the theoretical and semi-analytical models
should dominate the HI ionizing background at z~3. We aim at measuring the
Lyman continuum escape fraction, which is one of the key parameters to compute
the contribution of star-forming galaxies to the UV background. We have used
ultra-deep U-band imaging (U=30.2mag at 1sigma) by LBC/LBT in the
CANDELS/GOODS-North field, as well as deep imaging in COSMOS and EGS fields, in
order to estimate the Lyman continuum escape fraction of 69 star-forming
galaxies with secure spectroscopic redshifts at 3.27<z<3.40 to faint magnitude
limits (L=0.2L*, or equivalently M1500~-19). We have measured through stacks a
stringent upper limit (<1.7% at 1sigma) for the relative escape fraction of HI
ionizing photons from bright galaxies (L>L*), while for the faint population
(L=0.2L*) the limit to the escape fraction is ~10%. We have computed the
contribution of star-forming galaxies to the observed UV background at z~3 and
we have found that it is not enough to keep the Universe ionized at these
redshifts, unless their escape fraction increases significantly (>10%) at low
luminosities (M1500>-19). We compare our results on the Lyman continuum escape
fraction of high-z galaxies with recent estimates in the literature and discuss
future prospects to shed light on the end of the Dark Ages. In the future,
strong gravitational lensing will be fundamental to measure the Lyman continuum
escape fraction down to faint magnitudes (M1500~-16) which are inaccessible
with the present instrumentation on blank fields.
