Unique expansions of real numbers

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dc.creatorde Vries, Martijn
dc.creatorKomornik, Vilmos
dc.date2006-09-25
dc.date2008-12-18
dc.date.accessioned2026-07-07T12:16:58Z
dc.date.available2026-07-07T12:16:58Z
dc.descriptionIt was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i \in [0,q)$ satisfies the equality $\sum_{i=1}^\infty c_iq^{-i}=1$. 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 $\mathcal{U}_q$ of real numbers $x$ having a unique representation of the form $\sum_{i=1}^\infty c_iq^{-i}=x$ with integers $c_i$ 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 $\mathcal{U}_q$ is closed or even a Cantor set. We also study the set $\mathcal{U}_q'$ consisting of all sequences $(c_i)$ of integers $c_i \in [0,q)$ such that $\sum_{i=1}^{\infty} c_i q^{-i} \in \mathcal{U}_q$. We determine the numbers $r >1$ for which the map $q \mapsto \mathcal{U}_q'$ (defined on $(1, \infty)$) is constant in a neighborhood of $r$ and the numbers $q >1$ for which $\mathcal{U}_q'$ is a subshift or a subshift of finite type.
dc.description29 pages, some new results added, final version, to appear in Advances in Mathematics
dc.identifierhttps://arxiv.org/abs/math/0609708
dc.identifierhttp://arxiv.org/abs/math/0609708
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/211973
dc.subjectNumber Theory
dc.subjectCombinatorics
dc.subject11A63, 11B83, 37B10
dc.titleUnique expansions of real numbers
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