Generating normal numbers over Gaussian integers

Abstract

In this paper we consider normal numbers over Gaussian integers generated by polynomials. This corresponds to results by Nakai and Shiokawa in the case of real numbers. A generalization of normal numbers to Gaussian integers and their canonical number systems, which were characterized by Katai and Szabo, is given. With help of this we construct normal numbers of the form 0. bf(z1)c bf(z2)c bf(z3)c bf(z4)c bf(z5)c bf(z6)c . . . Here we denote by bxc the expansion of the integer part of x with respect to a given number system in Z[i], by f we denote a polynomial with complex coefficients, and by zi we denote a numbering of the Gaussian integers. We are able to show, that a number, which is constructed in this way, is normal to the given number system.