Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence a β-expansion for x if . We call a finite sequence an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes.
Given , we introduce the following subset of ℝ:
In other words, is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities
.
When , the Borel-Cantelli lemma tells us that the Lebesgue measure of is...