Expansions in non-integer bases: lower order revisited

Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(\varepsilon_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if $$x=\sum_{i=1}^{\infty}\varepsilon_iq^{-i}.$$ 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 $\min\mathcal B_2=q_2\approx 1.71064$, the appropriate root of $x^{4}=2x^{2}+x+1$. In this paper we show that for any $k\geq 3$, $\min\mathcal B_k=q_f\approx1.75488$, the appropriate root of $x^3=2x^2-x+1$.