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 x = ∑n = 1∞ an q- n with an ∈ {0, 1}. It was shown in [P. Erdo{combining double acute accent}s, I. Joó, V. Komornik, Characterization of the unique expansions 1 = ∑i = 1∞ q- ni and related problems, Bull. Soc. Math. France 118 (1990) 377-390] that if q <(sqrt(5) + 1) / 2, then each x ∈ (0, 1 / (q - 1)) has a continuum of such expansions; however, if q > (sqrt(5) + 1) / 2, 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 q <q2 = 1.71 ..., then each x has either 1 or infinitely many expansions, i.e., there are no such q in ((sqrt(5) + 1) / 2, q2). On the other hand, for each m > 1 there exists γm > 0 such that for any q ∈ (2 - γm, 2), there exists x which has exactly m expansions in base q. © 2009 Elsevier Inc. All rights reserved.