Periodic expansion of one by Salem numbers

We show that for a Salem number $\beta$ of degree $d$, there exists a positive constant $c(d)$ that ${\beta }^m$ is a Parry number for integers $m$ of natural density $\ge c(d)$. Further, we show $c(6)>1\mathrm{/}2$ and discuss a relation to the discretized rotation in dimension $4$.