Pseudo-randomness of a random Kronecker sequence: An instance of dynamical analysis

Abstract

This work focuses on the study of random properties of the celebrated Kronecker sequence K(a) formed of the fractional parts of the multiples of a real a. We consider here five parameters --two distances, covered space, the discrepancy and the Arnold constant-- that can be viewed as measures of pseudo-randomness. Our studies follow a probabilistic point of view and consider two main cases: the case of a random real a and the case of a random rational a. The two cases exhibit a strong parallelism. The most random Kronecker sequences are those associated with a whose digits in the continued fraction expansion are uniformly bounded. This is why we also study the case where a is randomly chosen among the reals with bounded digits by some M>1 and we consider the transition when M goes to infinity.