  I show that a Planck-scale deformation of the relativistic dispersion
relation, which has been independently considered in the quantum-gravity
literature, can explain the surprising results of three classes of experiments:
(1) observations of cosmic rays above the expected GZK limit, (2) observations
of multi-TeV photons from the BL Lac object Markarian 501, (3) studies of the
longitudinal development of the air showers produced by ultra-high-energy
hadronic particles. Experiments now in preparation, such as the ones planned
for the GLAST space telescope, will provide an independent test of this
solution of the three experimental paradoxes.
