We use a sample of 1669 QSOs ($r<20.15$, $3.6<z<4.0$) from the BOSS survey to study the intrinsic shape of their continuum and the Lyman continuum photon escape fraction (f$_{esc}$), estimated as the ratio between the observed flux and the expected intrinsic flux (corrected for the intergalactic medium absorption) in the wavelength range 865-885 AA rest-frame. Modelling the intrinsic QSO continuum shape with a power-law, $F_{lambda}proptolambda^{-gamma}$, we find a median $gamma=1.30$ (with a dispersion of $0.38$, no dependence on the redshift and a mild intrinsic luminosity dependence) and a mean f$_{esc}=0.75$ (independent of the QSO luminosity and/or redshift). The f$_{esc}$ distribution shows a peak around zero and a long tail of higher values, with a resulting dispersion of $0.7$. If we assume for the QSO continuum a double power-law shape (also compatible with the data) with a break located at $lambda_{rm br}=1000$ AA and a softening $Deltagamma=0.72 $ at wavelengths shorter than $lambda_{rm br}$, the mean f$_{esc}$ rises to $=0.82$. Combining our $gamma$ and f$_{esc}$ estimates with the observed evolution of the AGN luminosity function (LF) we compute the AGN contribution to the UV ionizing background (UVB) as a function of redshift. AGN brighter than one tenth of the characteristic luminosity of the LF are able to produce most of it up $zsim 3$, if the present sample is representative of their properties. At higher redshifts a contribution of the galaxy population is required. Assuming an escape fraction of Lyman continuum photons from galaxies between $5.5$ and $7.6%$, independent of the galaxy luminosity and/or redshift, a remarkably good fit to the observational UVB data up to $zsim 6$ is obtained. At lower redshift the extrapolation of our empirical estimate agrees well with recent UVB observations, dispelling the so-called Photon Underproduction Crisis.
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