Periodicity in radix systems with base $3$

In this paper, we examine the base $3$ number expansions in the ring of integers with digits $\{0, a, -b\}$, $a, b\in\mathbb{N}$ and $a \equiv b \pmod{3}$. Among others, we show that for arbitrary natural numbers $n$ and $k$, there is a system where $n$ has a periodic expansion with length $k$. We specify an infinite number of radix systems for a given valid signature.