Expansions in Non-integer Bases: Lower, Middle and Top Order

Let q ∈ (1, 2); it is known that each x ∈ [0, 1/(q− 1)] has an expansion of the form x = ∑ ∞ n=1 anq −n with an ∈ {0, 1}. It was shown in [3] that if q < ( √ 5 + 1)/2, then each x ∈ (0, 1/(q − 1)) has a continuum of such expansions; however, if q > ( √ 5 + 1)/2, then there exist infinitely many x having a unique expansion [4]. 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 (( √ 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.