On the cardinality of β-expansions of some numbers

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.