On the cardinality of β-expansions of some numbers

Author:

Zou Yuru1,Li Wenxia2,Lu Jian1

Affiliation:

1. College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, P. R. China

2. Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China

Abstract

Let [Formula: see text]. It is well known that every [Formula: see text] has a [Formula: see text]-expansion of the form [Formula: see text] with [Formula: see text], where [Formula: see text] denotes the largest integer not exceeding [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the sets of all [Formula: see text]-expansions of [Formula: see text] and the set of [Formula: see text]-prefixes of all [Formula: see text]-expansions of [Formula: see text], respectively. We show that [Formula: see text], [Formula: see text] and [Formula: see text] are equivalent under a certain finiteness condition.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

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