Expansions in non-integer bases: lower order revisited

Let $q\in (1,2)$ and $x\in [0,\frac{1}{q-1}]$. We say that a sequence $({\epsilon }_i{)}_{i=1}^{\mathrm{\infty }}\in \{0,1{\}}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if $$x=\sum _{i=1}^{\mathrm{\infty }}{\epsilon }_i{q}^{-i}\mathrm{.}$$ For any $k\in \mathbb{N}$, let ${\mathcal{B}}_k$ denote the set of $q$ such that there exists $x$ with exactly $k$ expansions in base $q$. In [12] it was shown that $\min\mathcal B_2=q_2\approx 1.71064$,theappropriaterootof$x^{4}=2x^{2}+x+1$. In this paper we show that for any $k\ge 3$, $\min {\mathcal{B}}_k=q_f\approx 1.75488$, the appropriate root of $x^3=2x^2-x+1$.