Radix and Pseudodigit Representations in Zⁿ

2010/02/22 by Eva Curry, Curry, Eva
Computer Science · Mathematics · #11A63 #15B36 #42C40 #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A63 #msc:15B36 #msc:42C40

paper · pdf · doi:10.48550/arxiv.1002.4016

13 pages; 2 figures; to be revised

arxiv created 2010/02/22 · openalex publication_date 2010/02/22 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 (A^N)s for some vector s (N >= 0 will vary). We call the vectors s "pseudodigits", and call this decomposition of Zⁿ a "pseudodigit representation".

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