On a problem of countable expansions

For a real number q∈(1,2)q∈(1,2) and x∈[0,1/(q−1)]x∈[0,1/(q−1)], the infinite sequence (di)(di) is called a q-expansion of x ifx=∑i=1∞diqi,di∈{0,1}for all i≥1. For m=1,2,⋯m=1,2,⋯ or ℵ0ℵ0 we denote by BmBm the set of q∈(1,2)q∈(1,2) such that there exists x∈[0,1/(q−1)]x∈[0,1/(q−1)] having exactly m different q-expansions. It was shown by Sidorov [18] that q2:=min⁡B2≈1.71064q2:=min⁡B2≈1.71064, and later asked by Baker [1] whether q2∈Bℵ0q2∈Bℵ0? In this paper we provide a negative answer to this question and conclude that Bℵ0Bℵ0 is not a closed set. In particular, we give a complete description of x∈[0,1/(q2−1)]x∈[0,1/(q2−1)] having exactly two different q2q2-expansions.