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

Abstract

Let q ∈ ( 1 , 2 ) ; it is known that each x ∈ [ 0 , 1 / ( q − 1 ) ] has an expansion of the form x = ∑ n = 1 ∞ a n q − n with a n ∈ { 0 , 1 } . It was shown in [P. Erdős, I. Joó, V. Komornik, Characterization of the unique expansions 1 = ∑ i = 1 ∞ q − n i and related problems, Bull. Soc. Math. France 118 (1990) 377–390] 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 [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 < q 2 = 1.71 … , then each x has either 1 or infinitely many expansions, i.e., there are no such q in ( ( 5 + 1 ) / 2 , q 2 ) . On the other hand, for each m > 1 there exists γ m > 0 such that for any q ∈ ( 2 − γ m , 2 ) , there exists x which has exactly m expansions in base q.