Expansions in non-integer bases: the top order and the tail
Nikita Sidorov · arXiv (Cornell University) · 2006
ABSTRACT. 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 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 1 there exists γm> 0 such that for any q ∈ (2 −γm, 2), there exists x which has exactly m expansions in base q. 1.