Univoque sets for real numbers

Fan Lü; Bo Tan; Jun Wu

Fan Lü ; Bo Tan ; Jun Wu

Fundamenta Mathematicae, Tome 227 (2014), p. 69-83 / Harvested from The Polish Digital Mathematics Library

Résumé

For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with $x_{i} \in 0, 1$ . We prove that for any x ∈ (0,1), (x) contains a sequence ${β_{k}}_{k \geq 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\bar{(x)}$ are nowhere dense.

Publié le : 2014-01-01

Zbl 06330487

EUDML-ID : urn:eudml:doc:283322

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     author = {Fan L\"u and Bo Tan and Jun Wu},
     title = {Univoque sets for real numbers},
     journal = {Fundamenta Mathematicae},
     volume = {227},
     year = {2014},
     pages = {69-83},
     zbl = {06330487},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-1-5}
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Fan Lü; Bo Tan; Jun Wu. Univoque sets for real numbers. Fundamenta Mathematicae, Tome 227 (2014) pp. 69-83. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-1-5/