  We prove that the set of growth rates of permutation classes includes an
infinite sequence of intervals whose infimum is $\theta_B\approx2.35526$, and
that it also contains every value at least $\lambda_B\approx2.35698$. These
results improve on a theorem of Vatter, who determined that there are
permutation classes of every growth rate at least $\lambda_A\approx2.48187$.
Thus, we also refute his conjecture that the set of growth rates below
$\lambda_A$ is nowhere dense. Our proof is based upon an analysis of expansions
of real numbers in non-integer bases, the study of which was initiated by
R\'enyi in the 1950s. In particular, we prove two generalisations of a result
of Pedicini concerning expansions in which the digits are drawn from sets of
allowed values.
