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

TL;DR

This paper investigates number systems with matrix bases similar to Jordan blocks, classifying digit sets for complete representations of integer vectors and analyzing the complexity of zero representations.

Contribution

It classifies digit sets for bases similar to Jordan blocks and analyzes the complexity of zero representations in these systems.

Findings

For $J_2$, digit sets of size 2 suffice for all $Z^2$.

For $J_n$ with $n extgreater 2$, three digits suffice for $Z^n$.

The zero representation language for $J_2$ with digits $(0, extpm 1)^T$ is not context-free, but Turing-recognizable.

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 $M=J_2$, we classify digit sets of size 2 allowing representation of the whole $\mathbb{Z}^2$. For $J_n$ with $n\geq 3$, it is shown that three digits suffice to represent all of $\mathbb{Z}^n$. 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 $M=J_2$ and the digits $(0,\pm 1)^T$ is shown not to be context-free, but to be recognizable by a Turing machine with logarithmic memory.