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 A Perfect Set of Reals with Finite Self-Information

2013 ◽  
Vol 78 (4) ◽  
pp. 1229-1246 ◽  
Author(s):  
Ian Herbert
Keyword(s):  
Hausdorff Dimension ◽  
Mutual Information ◽  
Kolmogorov Complexity ◽  
Packing Dimension ◽  
Specific Choice ◽  
Open Questions ◽  
Perfect Set ◽  
Definition Of ◽  
Image Position

AbstractWe examine a definition of the mutual information of two reals proposed by Levin in [5]. The mutual information iswhereK(·) is the prefix-free Kolmogorov complexity. A realAis said to have finite self-information ifI (A : A)is finite. We give a construction for a perfect Π10class of reals with this property, which settles some open questions posed by Hirschfeldt and Weber. The construction produces a perfect set of reals withK(σ)≤+KA(σ)+f (σ)for any given Δ20fwith a particularly nice approximation and for a specific choice of f it can also be used to produce a perfect Π10set of reals that are low for effective Hausdorff dimension and effective packing dimension. The construction can be further adapted to produce a single perfect set of reals that satisfyK(σ)≤+KA(σ)+f (σ)for allfin a ‘nice’ class of Δ20functions which includes all Δ20orders.

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 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.

Dimensions of measures under orthogonal projections

1996 ◽  
Vol 28 (2) ◽  
pp. 344-345
Author(s):  
Martina Zähle
Keyword(s):  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Packing Measure ◽  
Orthogonal Projections

Let dimH, E be the Hausdorff dimension and dimP, E the packing dimension of the subset E of ℝn given by the unique exponent where the corresponding Hausdorff or packing measure of E jumps from infinity to zero.

scholarly journals Hausdorff dimension and uniform exponents in dimension two

2018 ◽  
Vol 167 (02) ◽  
pp. 249-284 ◽  
Author(s):  
YANN BUGEAUD ◽  
YITWAH CHEUNG ◽  
NICOLAS CHEVALLIER
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Two Dimensional ◽  
Joint Work ◽  
Unpublished Work

AbstractIn this paper we prove that the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponentμin (1/2, 1) is equal to 2(1 −μ) forμ⩾$\sqrt2/2$, whereas forμ<$\sqrt2/2$it is greater than 2(1 −μ) and at most equal to (3 − 2μ)(1 − μ)/(1 −μ+μ2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) whenμtends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. Moreover, we prove a lower bound for the packing dimension, which appears to be strictly greater than the Hausdorff dimension for μ ⩾ 0.565. . . .

scholarly journals Some typical properties of dimensions of sets and measures

2005 ◽  
Vol 2005 (3) ◽  
pp. 239-254
Author(s):  
Józef Myjak
Keyword(s):  
Hausdorff Dimension ◽  
Correlation Dimension ◽  
Packing Dimension ◽  
Box Dimension ◽  
Local Dimension

This paper contains a review of recent results concerning typical properties of dimensions of sets and dimensions of measures. In particular, we are interested in the Hausdorff dimension, box dimension, and packing dimension of sets and in the Hausdorff dimension, box dimension, correlation dimension, concentration dimension, and local dimension of measures.

DIMENSION ANALYSIS OF CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1730001 ◽  
Author(s):  
JUN WANG ◽  
KUI YAO
Keyword(s):  
Hausdorff Dimension ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Packing Dimension ◽  
Box Dimension ◽  
Box Counting ◽  
Dimension Analysis ◽  
Box Counting Dimension ◽  
Self Similar

In this paper, we mainly discuss fractal dimensions of continuous functions with unbounded variation. First, we prove that Hausdorff dimension, Packing dimension and Modified Box-counting dimension of continuous functions containing one UV point are [Formula: see text]. The above conclusion still holds for continuous functions containing finite UV points. More generally, we show the result that Hausdorff dimension of continuous functions containing countable UV points is [Formula: see text] also. Finally, Box dimension of continuous functions containing countable UV points has been proved to be [Formula: see text] when [Formula: see text] is self-similar.

scholarly journals Resonance between Cantor sets

2009 ◽  
Vol 29 (1) ◽  
pp. 201-221 ◽  
Author(s):  
YUVAL PERES ◽  
PABLO SHMERKIN
Keyword(s):  
Hausdorff Dimension ◽  
Iterated Function System ◽  
Unit Interval ◽  
Irrational Rotation ◽  
Cantor Sets ◽  
Scaling Parameter ◽  
Iterated Function ◽  
Self Similar ◽  
Image Position ◽  
Geometric Resonance

AbstractLet Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

scholarly journals Relative multifractal box-dimensions

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2841-2859 ◽  
Author(s):  
Najmeddine Attia ◽  
Bilel Selmi
Keyword(s):  
Hausdorff Dimension ◽  
Multifractal Spectrum ◽  
Packing Dimension ◽  
Probability Measures ◽  
Multifractal Formalism ◽  
The Relationship ◽  
Box Dimensions

Given two probability measures ? and ? on Rn. We define the upper and lower relative multifractal box-dimensions of the measure ? with respect to the measure ? and investigate the relationship between the multifractal box-dimensions and the relative multifractal Hausdorff dimension, the relative multifractal pre-packing dimension. We also, calculate the relative multifractal spectrum and establish the validity of multifractal formalism. As an application, we study the behavior of projections of measures obeying to the relative multifractal formalism.

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