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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...
In a recent paper of Feng and Sidorov they show that for β ∈ (1,(1+√5)/2) the set of β-expansions grows exponentially for every x ∈ (0,1/(β-1)). In this paper we study this growth rate further. We also consider the set of β-expansions from a dimension theory perspective.
Let K ⊆ ℝ be the unique attractor of an iterated function system. We consider the case where K is an interval and study those elements of K with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence, we can show that the set of points with a unique coding is a graph-directed self-similar set in the sense of Mauldin and Williams (1988). The theory of Mauldin and Williams then provides a method...
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