  Beta-integers (``$\beta$-integers'') are those numbers which are the
counterparts of integers when real numbers are expressed in irrational basis
$\beta > 1$. In quasicrystalline studies $\beta$-integers supersede the
``crystallographic'' ordinary integers. When the number $\beta$ is a Parry
number, the corresponding $\beta$-integers realize only a finite number of
distances between consecutive elements and somewhat appear like ordinary
integers, mainly in an asymptotic sense. In this letter we make precise this
asymptotic behavior by proving four theorems concerning Parry $\beta$-integers.
