Non-Expansive Matrix Based number Systems
Abstract
An integer d-dimensional vector is usually represented by d strings, which express the value of each component separately. This article is devoted to number systems that allow representing integer vectors using a single string of digits. This system is given by an integer square matrix M and a finite set 𝒟 of symbols, called digits. We focus on full number systems, in which every vector x⃗ ∈ ℤn can be written in the form x⃗=∑i=0k−1 Mi di with di ∈ 𝒟 and thus represented by the string dk−1dk−2⋯ d1d0. In contrast to the very well-described case where the base M is expansive, we focus on systems in which M is similar to a Jordan block with the eigenvalue ± 1. We follow the work of Caldwell, Hare, and Vávra who initiated the study of such systems. First, we address the question of choosing a minimal sized digit set 𝒟. Then we describe optimal representations for vectors from ℤ2, both with respect to their length and the occurrence of non-zero digits.
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BibTeX
@misc{blazek2026nonexpansive,
title = {{Non-Expansive Matrix Based number Systems}},
author = {Adam Blažek and Kevin G. Hare and Edita Pelantová},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.04990},
eprint = {2605.04990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.04990},
}
RIS
TY - UNPB
TI - Non-Expansive Matrix Based number Systems
AU - Adam Blažek
AU - Kevin G. Hare
AU - Edita Pelantová
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2605.04990
UR - https://arxiv.org/abs/2605.04990
AB - An integer d-dimensional vector is usually represented by d strings, which express the value of each component separately. This article is devoted to number systems that allow representing integer vectors using a single string of digits. This system is given by an integer square matrix M and a finite set 𝒟 of symbols, called digits. We focus on full number systems, in which every vector x⃗ ∈ ℤ^n can be written in the form x⃗=∑_i=0^k−1 M^i d_i with d_i ∈ 𝒟 and thus represented by the string d_k−1d_k−2⋯ d_1d_0. In contrast to the very well-described case where the base M is expansive, we focus on systems in which M is similar to a Jordan block with the eigenvalue ± 1. We follow the work of Caldwell, Hare, and Vávra who initiated the study of such systems. First, we address the question of choosing a minimal sized digit set 𝒟. Then we describe optimal representations for vectors from ℤ^2, both with respect to their length and the occurrence of non-zero digits.
ER -