Finite systems are virtually linear

I would like to talk about the interplay between automata theory, p-adic analysis and real analysis. In the applied side, this has some connections to quantum theory, actually discreteness (=quantization) and causality imply DeBroglie waves; in mathematical terms, any DeBroglie wave e^{ia+b} can be represented by a real plot of a finite automaton, and no other smooth real function can be represented this way. This result also has some cryptography.