Intervals of permutation class growth rates

We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θ_B ≈ 2.35526, and that it also contains every value at least λ_B ≈ 2.35698. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λ_A ≈ 2.48187. Thus, we also refute his conjecture that the set of growth rates below λ_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 Rényi 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.