Let
$q\in (1,2)$
. A
$q$
-expansion of a number
$x$
in
$[0,1/(q-1)]$
is a sequence
$({\it\delta}_{i})_{i=1}^{\infty }\in \{0,1\}^{\mathbb{N}}$
satisfying
$$\begin{eqnarray}x=\mathop{\sum }_{i=1}^{\infty }\frac{{\it\delta}_{i}}{q^{i}}.\end{eqnarray}$$
Let
${\mathcal{B}}_{\aleph _{0}}$
denote the set of
$q$
for which there exists
$x$
with a countable number of
$q$
-expansions, and let
${\mathcal{B}}_{1,\aleph _{0}}$
denote the set of
$q$
for which
$1$
has a countable number of
$q$
-expansions. In Erdős
et al [On the uniqueness of the expansions
$1=\sum _{i=1}^{\infty }q^{-n_{i}}$
.
Acta Math. Hungar.
58 (1991), 333–342] it was shown that
$\min {\mathcal{B}}_{\aleph _{0}}=\min {\mathcal{B}}_{1,\aleph _{0}}=(1+\sqrt{5})/2$
, and in S. Baker [On small bases which admit countably many expansions.
J. Number Theory
147 (2015), 515–532] it was shown that
${\mathcal{B}}_{\aleph _{0}}\cap ((1+\sqrt{5})/2,q_{1}]=\{q_{1}\}$
, where
$q_{1}\,({\approx}1.64541)$
is the positive root of
$x^{6}-x^{4}-x^{3}-2x^{2}-x-1=0$
. In this paper we show that the second smallest point of
${\mathcal{B}}_{1,\aleph _{0}}$
is
$q_{3}\,({\approx}1.68042)$
, the positive root of
$x^{5}-x^{4}-x^{3}-x+1=0$
. En route to proving this result, we show that
${\mathcal{B}}_{\aleph _{0}}\cap (q_{1},q_{3}]=\{q_{2},q_{3}\}$
, where
$q_{2}\,({\approx}1.65462)$
is the positive root of
$x^{6}-2x^{4}-x^{3}-1=0$
.