Expansions in non-integer bases: Lower, middle and top orders

Let q∈(1,2); it is known that each x∈[0,1/(q−1)] has an expansion of the form with an∈{0,1}. It was shown in [P. Erdős, I. Joó, V. Komornik, Characterization of the unique expansions and related problems, Bull. Soc. Math. France 118 (1990) 377–390] that if , then each x∈(0,1/(q−1)) has a continuum of such expansions; however, if , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535–543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m>1 of expansions in base q. In particular, we show that if q1 there exists γm>0 such that for any q∈(2−γm,2), there exists x which has exactly m expansions in base q.