Radix and Pseudodigit Representations in Z^n

TL;DR

This paper introduces radix and pseudodigit representations for vectors in Z^n, providing conditions under which matrices produce such representations and exploring their structure and properties.

Contribution

It extends radix representation concepts to Z^n, introduces pseudodigit representations, and establishes conditions for matrices to generate these representations.

Findings

A sufficient condition for a matrix to produce a radix representation in Z^n.

Any expanding matrix can be associated with a pseudodigit representation of Z^n.

Partitioning Z^n enables radix representations up to translation by pseudodigits.

Abstract

We define radix representations for vectors in Z^n analogously with radix representations in Z, and give a sufficient condition for a matrix A:Z^n -> Z^n 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:Z^n -> Z^n will not be too far from yielding a radix representation, in that we can partition Z^n into a finite number of sets such that A yields a radix representation on each set up to translation by (A^N)s for some vector s (N >= 0 will vary). We call the vectors s "pseudodigits", and call this decomposition of Z^n a "pseudodigit representation".