  Applying the recently constructed analytic representation for the pp
scattering amplitudes, we present a study of p-air cross sections, with
comparison to the data from Extensive Air Shower (EAS) measurements. The
amplitudes describe with precision all available accelerator data at ISR, SPS
and LHC energies, and its theoretical basis, together with the very smooth
energy dependence of parameters controlled by unitarity and dispersion
relations, permit reliable extrapolation to higher energies and to asymptotic
ranges. The comparison with cosmic ray data is very satisfactory in the whole
pp energy interval from 1 to 100 TeV. High energy asymptotic behaviour of cross
sections is investigated in view of the geometric scaling property of the
amplitudes. The amplitudes predict that the proton does not behave as a black
disk even at asymptotically high enegies, and we discuss possible non-trivial
consequences of this fact for pA collision cross sections at higher energies.
