On positional representation of integer vectors

From positional representation of numbers to positional representation of vectors

“…Any matrix numeration system used in this paper is given by a non-singular matrix base M ∈ Z m×m and a finite (vector) digit set D ⊂ Z m . Thanks to the result of [15] mentioned earlier, we always assume that Z m is a subset of Fin D (M) and,…”

“…Theorem 9 states that each non-singular matrix M ∈ Z m×m with no eigenvalue on the unit circle can be equipped with a suitable finite digit set D ⊂ Z m such that any integer vector x ∈ Z m is representable in the system (M, D). As shown in [15], the assumption |λ| ̸ = 1 for each eigenvalue λ of M is not necessary for the representability of Z m in (M, D). Nevertheless, this assumption is necessary for parallel addition in (M, D), as shown below.…”

“…Another generalisation introduced recently allows to use both positive and negative powers of the matrix base for representation of vectors. In [15], it is shown that for M ∈ Z m×m with det M = ∆ ̸ = 0, there exists a finite digit set D ⊂ Z m such that every integer vector from Z m has a finite (M, D)-representation, i.e.,…”