  Let ${\cal H}=(q_1, \ldots q_r)$ be a finite set of coprime integers and let
$n_1, n_2, \ldots$ denote the multiplicative semigroup generated by $\cal H$
and arranged in increasing order. The distribution of such sequences has been
studied intensively in number theory and they have remarkable probabilistic and
ergodic properties. For example, the asymptotic properties of the sequence
$\{n_kx\}$ are very similar to those of independent, identically distributed
random variables; here $\{\cdot \}$ denotes fractional part. However, the
behavior of this sequence depends sensitively on the generating elements of
$(n_k)$ and the combination of probabilistic and number-theoretic effects
results in a unique, highly interesting asymptotic behavior. In particular, the
properties of $\{n_kx\}$ are not permutation invariant, in contrast to i.i.d.
behavior. The purpose of this paper is to show that $\{n_kx\}$ satisfies a
strong independence property ("interlaced mixing"), enabling one to determine
the precise asymptotic behavior of permuted sums $S_N (\sigma)= \sum_{k=1}^N
f(n_{\sigma(k)} x)$. As we will see, the behavior of $S_N(\sigma)$ still
follows that of sums of independent random variables, but its growth speed
(depending on $\sigma$) is given by the classical G\'al function of Diophantine
approximation theory. Some examples describing the class of possible growth
functions are given.
