Expansions in non-integer bases: lower order revisited

Abstract

Let $q\in (1,2)$ and $x\in [0,\frac{1}{q-1}]$. We say that a sequence $({\epsilon }_i{)}_{i=1}^{\infty }\in \{0,1{\}}^{\mathbb{N}}$ 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 $k\ge 3$, $\min {\mathcal{B}}_k=q_f\approx 1.75488$, the appropriate root of $x^3=2x^2-x+1$.

16 pages