Univoque sets for real numbers

Fan Lü, Bo Tan, and Jun Wu. "Univoque sets for real numbers." Fundamenta Mathematicae 227.1 (2014): 69-83. <http://eudml.org/doc/283322>.

@article{FanLü2014,
abstract = {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 $x_\{i\} ∈ \{0,1\}$. We prove that for any x ∈ (0,1), (x) contains a sequence $\{β_\{k\}\}_\{k ≥ 1\}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\overline\{(x)\}$ are nowhere dense.},
author = {Fan Lü, Bo Tan, Jun Wu},
journal = {Fundamenta Mathematicae},
keywords = {univoque set; quasi-greedy expansion; Lebesgue measure; hausdorff dimension},
language = {eng},
number = {1},
pages = {69-83},
title = {Univoque sets for real numbers},
url = {http://eudml.org/doc/283322},
volume = {227},
year = {2014},
}

TY - JOUR
AU - Fan Lü
AU - Bo Tan
AU - Jun Wu
TI - Univoque sets for real numbers
JO - Fundamenta Mathematicae
PY - 2014
VL - 227
IS - 1
SP - 69
EP - 83
AB - 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 $x_{i} ∈ {0,1}$. We prove that for any x ∈ (0,1), (x) contains a sequence ${β_{k}}_{k ≥ 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\overline{(x)}$ are nowhere dense.
LA - eng
KW - univoque set; quasi-greedy expansion; Lebesgue measure; hausdorff dimension
UR - http://eudml.org/doc/283322
ER -