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) = c 0 + c 1 x + · · · + c k x k ∈ Z[x], c k = 1 satises the condition |c 0 | > 2 k i=1 |c i | then there is a suitable digit set D for which (Z k , M, D) is a number system, where M is the companion matrix of f (x).
Statements and earlier resultsA lattice in R k is the set of all integer combinations of k linearly independent vectors. It can be viewed either as a set of points in the k-dimensional Euclidean space, as a Z-module, or as a nitely generated free Abelian group.
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