On the cardinality of β-expansions of some numbers

Let β > 1. It is well known that every x [0,β/(β-1)] has a β-expansion of the form x =k=1∞ kβ-k with i {0, 1,..,β}, where β denotes the largest integer not exceeding β. Let ∑β(x) and ∑β,n(x) denote the sets of all β-expansions of x and the set of n-prefixes of all β-expansions of x, respectively. We show that ∑β(x) = 20, dimH∑β(x) > 0 and limn∞1 nlog ∑β,n(x) > 0 are equivalent under a certain finiteness condition.