Radix and Pseudodigit Representations in Zⁿ

الأصل · EN

We define radix representations for vectors in Zⁿ analogously with radix representations in Z, and give a sufficient condition for a matrix A:Zⁿ -> Zⁿ to yield a radix representation with a given canonical digit set. We relate our results to a sufficient condition given recently by Jeong. We also show that any expanding matrix A:Zⁿ -> Zⁿ will not be too far from yielding a radix representation, in that we can partition Zⁿ into a finite number of sets such that A yields a radix representation on each set up to translation by (Aⁿ)s for some vector s (N >= 0 will vary). We call the vectors s "pseudodigits", and call this decomposition of Zⁿ a "pseudodigit representation".