Generalized Hausdorff Dimensions of Networks

2020 ◽  
pp. 391-411
Author(s):  
Eric Rosenberg
Keyword(s):  
Hausdorff Dimensions

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.

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

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

On Hausdorff dimension monotonicity of a family of dynamical subsets of Rauzy fractals

2015 ◽  
Vol 11 (04) ◽  
pp. 1089-1098 ◽  
Author(s):  
W. Georg Nowak ◽  
Klaus Scheicher ◽  
Víctor F. Sirvent
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Dimensions ◽  
The Third

We consider a family of dynamically defined subsets of Rauzy fractals in the plane. These sets were introduced in the context of the study of symmetries of Rauzy fractals. We prove that their Hausdorff dimensions form an ultimately increasing sequence of numbers converging to 2. These results answer a question stated by the third author in 2012.

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