Non-expansive matrix number systems with bases similar to certain Jordan blocks

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 equal or similar to $J_n$, the Jordan block with eigenvalue 1 and dimension n. If $M=J_2$, we classify all digit sets of size two allowing representation for all of $Z^2$. For $M=J_n$ with $n\ge 3$, we show that a digit set of size three suffice to represent all of $Z^n$. For bases M similar to $J_n$, $n\ge 2$, we construct a digit set of size n such that all of $Z^n$ is represented. The language of words 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.