ASYMPTOTIC NORMALITY OF b-ADDITIVE FUNCTIONS ON POLYNOMIAL SEQUENCES IN THE GAUSSIAN NUMBER FIELD

0. Notations Throughout the paper we use the following notations: We write e(z) = e; C, R, Q, Z, N and N0, denote the set of complex numbers, real numbers, rational numbers, integers, positive integers, and positive integers including zero, respectively. Q(i) denotes the field of Gaussian numbers, and Z[i] the ring of Gaussian integers. We write tr(z) and N(z) for the trace and the norm of z over Q, and {z} for the minimal distance of a real number z to the next integer. Furthermore, the largest integer less than or equal to a real number z is denoted by [z]. λn denotes the n-dimensional Lebesgue measure. V T denotes the transposition of the matrix V . For a set A we denote its closure by Ā and its boundary by ∂A. Furthermore we use the symbol f g to mean that f = O (g) and f g to mean that g = O (f).