Journal of Physics A. Mathematical and General vol:38 issue:12 pages:2599-2622
We consider higher-dimensional generalizations of the classical one-dimensional 2-automatic paperfolding and Rudin-Shapiro, sequences on N. This is done by considering the same automaton-structure as in the one-dimensional case, but using binary number systems in Z(m) instead of in N. The correlation function and the diffraction spectrum for the resulting m-dimensional paperfolding and Rudin-Shapiro point sets are calculated through the corresponding sequences with values +/- 1. They are shown to be quasi-independent of the dimension m and of the particular binary number system under consideration. It is shown that any paperfolding sequence thus obtained has a discrete spectrum. The Rudin-Shapiro sequences have an absolutely continuous Lebesgue spectral measure.