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 Jn, the Jordan block with eigenvalue 1 and dimension n. If M=J2, we classify all digit sets of size two allowing representation for all of Z2. For M=Jn with n≥3, we show that a digit set of size three suffice to represent all of Zn. For bases M similar to Jn, n≥2, we construct a digit set of size n such that all of Zn is represented. The language of words 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.