Equidistribution of Matrix-Power Residues Modulo One
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Here {y\ = y — [y] = the fractional part of y. The number N is an integer >1. The number x0 is a given initial value such that 0 ^ Xo < 1. The number 0 is fixed. Some early references to numerical work with sequences of the type (1) are given by 0. Taussky and J. Todd in [1]. Regarding the sequence x„ as a function of Xo, I proved in [2] that for almost all x0 the sequence x„ is equidistributed modulo 1, i.e.,
[1] F. Riesz. Sur la théorie ergodique , 1944 .
[2] J. Franklin. Deterministic Simulation of Random Processes , 1963 .
[3] J. Franklin. On the equidistribution of pseudo-random numbers , 1958 .
[4] O. Taussky,et al. Generation and testing of pseudo-random numbers , 1954 .
[5] H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins , 1916 .