  We use ultra-deep ultraviolet VLT/VIMOS intermediate-band and VLT/FORS1
narrow-band imaging in the GOODS Southern field to derive limits on the
distribution of the escape fraction (f_esc) of ionizing radiation for L >~
L*(z=3) Lyman Break Galaxies (LBGs) at redshift 3.4--4.5. Only one LBG, at
redshift z=3.795, is detected in its Lyman continuum (LyC; S/N~5.5), the
highest redshift galaxy currently known with a direct detection. Its
ultraviolet morphology is quite compact (R_eff=0.8, kpc physical). Three out of
seven AGN are also detected in their LyC, including one at redshift z=3.951 and
z850 = 26.1. From stacked data (LBGs) we set an upper limit to the average
f_esc in the range 5%--20%, depending on the how the data are selected (e.g.,
by magnitude and/or redshift). We undertake extensive Monte Carlo simulations
that take into account intergalactic attenuation, stellar population synthesis
models, dust extinction and photometric noise in order to explore the moments
of the distribution of the escaping radiation. Various distributions
(exponential, log-normal and Gaussian) are explored. We find that the median
f_esc is lower than ~6% with an 84% percentile limit not larger than 20%. If
this result remains valid for fainter LBGs down to current observational
limits, then the LBG population might be not sufficient to account for the
entire photoionization budget at the redshifts considered here, with the exact
details dependent upon the assumed ionizing background and QSO contribution
thereto. It is possible that f_esc depends on the UV luminosity of the
galaxies, with fainter galaxies having higher f_esc, and estimates of f_esc
from a sample of faint LBG from the HUDF (i775<28.5) are in broad quantitative
agreement with such a scenario.
