Correlation and spectral properties of multidimensional Thue-Morse sequences

This paper considers higher-dimensional generalizations of the classical one-dimensional two-automatic Thue-Morse sequence on N. This is done by taking the same automaton-structure as in the one-dimensional case, but using binary number systems in Z(m) instead of in N. It is shown that the corresponding +/-1-valued Thue-Morse sequences are either periodic or have a singular continuous spectrum, dependent on the binary number system. Specific results are given for dimensions up to six, with extensive illustrations for the one-, two- and three-dimensional case.