Block additive functions on the Gaussian integers
نویسندگان
چکیده
We present three conceptually different methods to prove distribution results for block additive functions with respect to radix expansions of the Gaussian integers. Based on generating function approaches we obtain a central limit theorem and asymptotic expansions for the moments. Furthermore, these generating functions as well as ergodic skew products are used to prove uniform distribution in residue classes and modulo 1.
منابع مشابه
ASYMPTOTIC NORMALITY OF b-ADDITIVE FUNCTIONS ON POLYNOMIAL SEQUENCES IN CANONICAL NUMBER SYSTEMS
We consider b-additive functions f where b is the base of a canonical number system in an algebraic number field. In particular, we show that the asymptotic distribution of f(p(z)) with p a polynomial running through the integers of the algebraic number field is the normal law. This is a generalization of results of Bassily and Katai (for the integer case) and of Gittenberger and Thuswaldner (f...
متن کاملASYMPTOTIC 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 ov...
متن کاملASYMPTOTIC NORMALITY OF b-ADDITIVE FUNCTIONS ON POLYNOMIAL SEQUENCES IN THE GAUSSIAN NUMBER FIELD1
We consider the asymptotic behavior of b-additive functions f with respect to a base b of a canonical number system in the Gaussian number eld. In particular, we get a normal limit law for f(P (z)) where P (z) is a polynomial with integer coeecients. Our methods are exponential sums over the Gaussian number eld as well as certain results from the theory of uniform distribution. Throughout the p...
متن کاملBlock-Matching Sub-Pixel Motion Estimation from Noisy, Under-Sampled Frames — An Empirical Performance Evaluation
The performance of block-matching sub-pixel motion estimation algorithms under the adverse conditions of image undersampling and additive noise is studied empirically. This study is motivated by the requirement for reliable subpixel accuracy motion estimates for motion compensated observation models used in multi-frame super-resolution image reconstruction. Idealized test functions which includ...
متن کاملA Structure Theorem for Multiplicative Functions over the Gaussian Integers and Applications
We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U3-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadrat...
متن کامل