For $\mathbf{r}=(r_{1},\ldots,r_{d}) \in\mathbb{R}^{d}$ the
map $\tau_{\mathbf{r}}\colon \mathbb{Z}^{d} \to \mathbb{Z}^{d}$
given by
\[ \tau_{\mathbf{r}}(a_{1},\ldots,a_{d})
=(a_{2},\ldots,a_{d},-\lfloor r_{1}a_{1}+ \cdots + r_{d}a_{d}
\rfloor) \]
is called a shift
radix system if for each $\mathbf{a} \in \mathbb{Z}^{d}$ there
exists an integer $k>0$ with $\tau_{\mathbf{r}}^{k}(\mathbf{a})=0$.
As shown in the first two parts of this series of papers shift
radix systems are intimately related to certain well-known
notions of number systems like $\beta$-expansions and canonical
number systems. In the present paper further structural relationships
between shift radix systems and canonical number systems are
investigated. Among other results we show that canonical number
systems related to polynomials
\[ \sum_{i=0}^{d} p_{i} X^{i} \in \mathbb{Z}[X] \]
of degree $d$ with a large but
fixed constant term $p_{0}$ approximate the set of ($d-1$)-dimensional shift radix systems. The proofs make extensive
use of the following tools: Firstly, vectors $\mathbf{r} \in\mathbb{R}^{d}$
which define shift radix systems are strongly connected to
monic real polynomials all of whose roots lie inside the unit
circle. Secondly, geometric considerations which were established
in Part I of this series of papers are exploited. The main
results establish two conjectures mentioned in Part II of
this series of papers.