The study of the interactions of Cosmic Rays (CR's) with universal diffuse
background radiation can provide very stringent tests of the validity of
Special Relativity. The interactions we consider are the ones characterized by
well defined energy thresholds whose energy position can be predicted on the
basis of Special Relativity. We argue that the experimental confirmation of the
existence of these thresholds can in principle put very stringent limits on the
scale where Special Relativity and/or continuity of space-time may possibly
break down.Comment: 20 pages, 3 tables, no figures. Version to appear in Phys. Rev. D. A
few changes in the presentation. Conclusions unchanged. Some references adde
