Expansions in non-integer bases: lower, middle and top orders

Let $q\in(1,2)$; it is known that each $x\in[0,1/(q-1)]$ has an expansion of the form $x=\sum_{n=1}^\infty a_nq^{-n}$ with $a_n\in\{0,1\}$. It was shown in \cite{EJK} that if $q<(\sqrt5+1)/2$, then each $x\in(0,1/(q-1))$ has a continuum of such expansions; however, if $q>(\sqrt5+1)/2$, then there exist infinitely many $x$ having a unique expansion \cite{GS}. In the present paper we begin the study of parameters $q$ for which there exists $x$ having a fixed finite number $m>1$ of expansions in base $q$. In particular, we show that if $q<q_2=1.71...$, then each $x$ has either 1 or infinitely many expansions, i.e., there are no such $q$ in $((\sqrt5+1)/2,q_2)$. On the other hand, for each $m>1$ there exists $\ga_m>0$ such that for any $q\in(2-\ga_m,2)$, there exists $x$ which has exactly $m$ expansions in base $q$.