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

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.

Hausdorff dimension and distance sets

1994 ◽  
Vol 87 (1-3) ◽  
pp. 193-201 ◽  
Author(s):  
Jean Bourgain
Keyword(s):  
Hausdorff Dimension ◽  
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.

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