On number system constructions

We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M−1) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M−1. We shall prove further that if the polynomial f(x) = c0 + c1x + ··· + ckxk ∈ Z[x], ck = 1 satisfies the condition |c0| > 2 Σi=1k |ci| then there is a suitable digit set D for which (Zk, M, D) is a number system, where M is the companion matrix of f(x).