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Non-expansive matrix number systems with bases similar to $J_n(1)$ (2110.11937v1)
Published 22 Oct 2021 in math.NT
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.