Hausdorff dimensions of sets related to Lüroth expansion

2016 ◽  
Vol 150 (2) ◽  
pp. 286-302 ◽  
Author(s):  
Y. Gui ◽  
W. Li
Keyword(s):  
Hausdorff Dimensions ◽  
Lüroth Expansion

THE GROWTH RATE OF THE DIGITS IN THE LÜROTH EXPANSIONS

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050064
Author(s):  
MEIYING LÜ
Keyword(s):  
Growth Rate ◽  
Hausdorff Dimension ◽  
Exponential Rate ◽  
Hausdorff Dimensions ◽  
Lüroth Expansion

For any [Formula: see text], let [Formula: see text] be its Lüroth expansion with digits [Formula: see text]. This paper is concerned with the growth rate of the digits in the Lüroth expansions. Let [Formula: see text] be a function satisfying [Formula: see text] as [Formula: see text] and [Formula: see text]. In this paper, we consider the set [Formula: see text] and we quantify the size of [Formula: see text] in the sense of Hausdorff dimension. As applications, we get the Hausdorff dimensions of the sets of points for which [Formula: see text] grows with polynomial and exponential rate.

scholarly journals The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

2020 ◽  
Vol 378 (1) ◽  
pp. 625-689 ◽  
Author(s):  
Ewain Gwynne
Keyword(s):  
Hausdorff Dimension ◽  
Quantum Gravity ◽  
Free Field ◽  
Gaussian Free Field ◽  
Hausdorff Dimensions ◽  
Thick Points

Abstract Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

Exceptional sets related to the largest digits in Lüroth expansions

2021 ◽  
pp. 1-15
Author(s):  
Shuyi Lin ◽  
Jinjun Li ◽  
Manli Lou
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Level Sets ◽  
Disjoint Union ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Lüroth Expansion

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

On the Hausdorff dimensions of distance sets

Mathematika ◽  
1985 ◽  
Vol 32 (2) ◽  
pp. 206-212 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Hausdorff Dimensions ◽  
Distance Sets

Relative convergence speed of Lüroth expansions and Hausdorff dimension

2021 ◽  
pp. 1-21
Author(s):  
Xiaoyan Tan ◽  
Jia Liu ◽  
Zhenliang Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Irrational Number ◽  
Convergence Speed ◽  
Rate Of Growth ◽  
Dimension Formula ◽  
Lüroth Expansion

For any [Formula: see text] in [Formula: see text], let [Formula: see text] be the Lüroth expansion of [Formula: see text]. In this paper, we study the relative convergence speed of its convergents [Formula: see text] to the rate of growth of digits in the Lüroth expansion of an irrational number. For any [Formula: see text] in [Formula: see text], the sets [Formula: see text] and [Formula: see text] are proved to be of same Hausdorff dimension [Formula: see text]. Furthermore, for any [Formula: see text] in [Formula: see text] with [Formula: see text], the Hausdorff dimension of the set [Formula: see text] [Formula: see text] is proved to be either [Formula: see text] or [Formula: see text] according as [Formula: see text] or not.

Dimension spectra of lines1

Computability ◽  
2021 ◽  
pp. 1-28
Author(s):  
Neil Lutz ◽  
D.M. Stull
Keyword(s):  
Hausdorff Dimension ◽  
Kolmogorov Complexity ◽  
Euclidean Plane ◽  
Unit Interval ◽  
Packing Dimension ◽  
Hausdorff Dimensions ◽  
Dimension Spectrum ◽  
Algorithmic Dimension

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

scholarly journals Estimating the Hausdorff dimensions of univoque sets for self-similar sets

2019 ◽  
Vol 30 (5) ◽  
pp. 862-873 ◽  
Author(s):  
Xiu Chen ◽  
Kan Jiang ◽  
Wenxia Li
Keyword(s):  
Hausdorff Dimensions ◽  
Self Similar

scholarly journals Hausdorff dimensions in p-adic analytic groups

2019 ◽  
Vol 231 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Benjamin Klopsch ◽  
Anitha Thillaisundaram ◽  
Amaia Zugadi-Reizabal
Keyword(s):  
Hausdorff Dimensions
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