Generalized number systems in Euclidean spaces

The concept of number systems in higher-dimensional Euclidean spaces as well as in number fields are introduced and treated. The paper is mainly a survey of known results. If ofZk denotes the ring of integer vectorials in Euclidean space Rk then we study the subgroup L=MZk where M is a k × k matrix with integer entries. A digit set is defined with the help of residue classes (mod L)Aj (j = 0,…, t − 1), where t = |det M|. The analogous of q-additive and q-multiplicative functions over the ring of Gaussian integers are also given.