Unique expansions of real numbers
By Martijn de Vries, Vilmos Komornik. 2006
157 citations recorded by the source corpus.
It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality ∑i=1∞ciq−i=1. The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form ∑i=1∞ciq−i=x with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set Uq′ consisting of all sequences (ci) of integers ci∈[0,q) such that ∑i=1∞ciq−i∈Uq. We determine the numbers r>1 for which the map q↦Uq′ (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which Uq′ is a subshift or a subshift of finite type.
Subfields: Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6ef874888db89b26c17bd7220e8676caeda38bc2
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In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Hausdorff dimension of univoque sets and Devil's staircase (2015, background): “ows from this theorem thatUq is a(Lebesgue) null set forall q 6= M +1, while UM+1 ⊆ [0,1] has measure one because its complementer set is countable in [0,1]. Since Uq \Uq is countable for each q (see [8]), the same properties hold for Uq as well. Theorem 1.2. For almost all q > 1, U′ q is a subsh”