 
Summary: From number systems to shift radix systems
Shigeki Akiyama
and Klaus Scheicher
Abstract
Shift radix systems provide a unified notion to study two important
types of number systems. In this paper, we briefly review the origin of this
notion.
1. Introduction
Let r = (r1, . . . , rd) Rd
. Consider a mapping r : Zd
Zd
, which maps each
element (z1, . . . , zd) to (z2, . . . , zd+1), provided that
0 r1z1 + r2z2 + · · · + rdzd + zd+1 < 1.
Obviously, r is defined by
r (z1, . . . , zd) = (z2, . . . , zd,  r1z1 + · · · + rdzd ). (1.1)
We say that r has the finiteness property if for every z Zd
there exists a k,
such that k
r (z) = 0.
