Non-expansive matrix number systems with bases similar to J_n(1)

2021/10/22 by Joshua W. Caldwell, Caldwell, Joshua W., Kevin G. Hare +3
Computer Science · #11C20 #11K16 #15B36 #Algorithms and Data Compression #Cellular Automata and Applications #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2110.11937

openalex publication_date 2021/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study representations of integral vectors in a number system with a matrix base M and vector digits. We focus on the case when M is similar to J_n, the Jordan block of 1 of size n. If M=J₂, we classify digit sets of size 2 allowing representation of the whole ℤ². For J_n with n≥ 3, it is shown that three digits suffice to represent all of ℤⁿ. For bases similar to J_n, at most n digits are required, with the exception of n=1. Moreover, the language of strings representing the zero vector with M=J₂ and the digits (0,± 1)^T is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.

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