 
Summary: Regular maps in generalized number systems
J.P. Allouche, K. Scheicher and R. F. Tichy
Abstract
This paper extends some results of Allouche and Shallit for qregular
sequences to numeration systems in algebraic number fields and to linear
numeration systems. We also construct automata that perform addition
and multiplication by a fixed number.
1 Introduction
A sequence is called qautomatic if its nth term can be generated by a finite
state machine from the qary digits of n. The concept of automatic sequences
was introduced in 1969 and 1972 by Cobham [8, 9]. In 1979 Christol [6] (see also
Christol, Kamae, Mend`es France and Rauzy [7]) discovered a nice arithmetic
property of automatic sequences: a sequence with values in a finite field of
characteristic p is pautomatic if and only if the corresponding power series is
algebraic over the field of rational functions over this finite field. A brief survey
on this subject is given in [2], see also [10]. Some generalizations of this concept
were studied in [27, 23, 24, 3], see also the survey [1]. An automatic sequence
has to take its values in a finite set. To relax this condition, Allouche and
Shallit [5] introduced the notion of qregular sequences. To give a hint of what
qregularity is, let us consider the following example. If S(n) is the sum of the
