## On number system constructions.(English)Zbl 1153.11003

Let $$\Lambda$$ be a lattice, i.e., a free $${\mathbb Z}$$-module of rank $$k$$, in $${\mathbb R}^k$$, $$M:\Lambda\to\Lambda$$ a group endomorphism with $$\det(M)\neq 0$$ and $$D$$ a finite subset of $$\Lambda$$ containing $$0$$. The triple $$(\Lambda,M,D)$$ is called a number system, if every element $$x\in\Lambda$$ has a unique, finite representation of the form $$x=\sum_{i=0}^{\ell} M^i d_i$$, where $$d_i\in D$$ and $$\ell\in{\mathbb N}$$.
The first half of the paper is devoted to an overview on results in this area. In the second part, the authors prove two results.
Let $$\Lambda$$ and $$M$$ be given. If the spectral radius of $$M^{-1}$$ is less than $$1/2$$, then there exists a digit set $$D$$ for which $$(\Lambda, M, D)$$ is a number system.
Let $$f=c_0+c_1x+\cdots +c_k x^k\in{\mathbb Z}[x]$$ with $$c_k=1$$ be a polynomial and $$M$$ its companion matrix. If the weak dominant condition $$| c_0| >2\sum_{i=1}^k | c_i|$$ holds, then there exists a digit set $$D$$ for which $$({\mathbb Z}^k, M, D)$$ is a number system.