Osculatory Packings by Spheres

1970 ◽  
Vol 13 (1) ◽  
pp. 59-64 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Simple Proof ◽  
Pairwise Disjoint ◽  
Exponent Of Convergence ◽  
Residual Set ◽  
Open Set ◽  
Finite Lebesgue Measure

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

scholarly journals Relative bifurcation sets and the local dimension of univoque bases

2020 ◽  
pp. 1-33
Author(s):  
PIETER ALLAART ◽  
DERONG KONG
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Expression ◽  
Lebesgue Measure ◽  
Pairwise Disjoint ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Local Dimension ◽  
Dimension Zero ◽  
Image Position ◽  
Dimension One

Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$ . The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit $$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$ for all $q\in (1,M+1)$ . We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$ , where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.

scholarly journals On two functionals involving the maximum of the torsion function

2018 ◽  
Vol 24 (4) ◽  
pp. 1585-1604 ◽  
Author(s):  
Antoine Henrot ◽  
Ilaria Lucardesi ◽  
Gérard Philippin
Keyword(s):  
Lebesgue Measure ◽  
Lower Bounds ◽  
Convex Sets ◽  
Upper And Lower Bounds ◽  
Dirichlet Eigenvalue ◽  
Torsion Function ◽  
Open Set ◽  
First Dirichlet Eigenvalue ◽  
Finite Lebesgue Measure

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider T(Ω)∕(M(Ω)|Ω|) and M(Ω)λ1(Ω), where Ω is a bounded open set of ℝd with finite Lebesgue measure |Ω|, M(Ω) denotes the maximum of the torsion function, T(Ω) the torsion, and λ1(Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

scholarly journals NONSTANDARD ANALYSIS, FRACTAL PROPERTIES AND BROWNIAN MOTION

Fractals ◽  
2009 ◽  
Vol 17 (01) ◽  
pp. 117-129 ◽  
Author(s):  
PAUL POTGIETER
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Simple Proof ◽  
Nonstandard Analysis ◽  
Counting Measure ◽  
And Brownian Motion ◽  
Fractal Properties

In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman's lemma and find that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a simple proof of the doubling of the dimension of certain sets under a Brownian motion.

scholarly journals On Efficiency and Localisation for the Torsion Function

2021 ◽  
Author(s):  
M. van den Berg ◽  
D. Bucur ◽  
T. Kappeler
Keyword(s):  
Lebesgue Measure ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Dirichlet Laplacian ◽  
Torsion Function ◽  
Open Set ◽  
Finite Lebesgue Measure

AbstractWe consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set ${\Omega }\subset \mathbb {R}^{m}$ Ω ⊂ ℝ m of finite Lebesgue measure $0<|{\Omega }|<\infty $ 0 < | Ω | < ∞ with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

scholarly journals Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

2021 ◽  
Author(s):  
Jiahao Qiu ◽  
Jianjie Zhao
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Minimal System ◽  
Combinatorial Properties ◽  
Return Times ◽  
Residual Set ◽  
Closed Sets ◽  
Open Set ◽  
Geometric Progressions ◽  
Set Of Points

AbstractIn this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

A new proof for the residual set dimension of the apollonian packing

1984 ◽  
Vol 96 (3) ◽  
pp. 413-423 ◽  
Author(s):  
Claude Tricot
Keyword(s):  
Hausdorff Dimension ◽  
Density Theorem ◽  
Residual Set ◽  
Curvilinear Triangle

AbstractLet (Dn) be the apollonian packing of a curvilinear triangle T, ρn the radius of Dn, E = T—U Dn the residual set, dim (E) its Hausdorff dimension. In this paper we give a new proof of the equality dim proved by Boyd [2]. Our technique is to construct a sequence of regular triangles covering E, and suitable measures μkcarried by E which allow us to apply a density theorem.

scholarly journals The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping

2002 ◽  
Vol 31 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Hongwen Guo ◽  
Dihe Hu
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Intersection Property ◽  
Random Sets ◽  
Hausdorff Measures ◽  
Open Set Condition ◽  
Finite Intersection ◽  
Open Set ◽  
Finite Intersection Property

We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.

scholarly journals Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms

1989 ◽  
Vol 9 (3) ◽  
pp. 403-425 ◽  
Author(s):  
Lorenzo J. Diaz ◽  
Marcelo Viana
Keyword(s):  
Hausdorff Dimension ◽  
Equilibrium Measures ◽  
Open Set ◽  
Limit Capacity ◽  
Nonwandering Set

AbstractWe consider one-parameter families of torus diffeomorphisms that bifurcate from global hyperbolic maps (Anosov) to DA maps (derived from Anosov). For an open set of these families, we show that the Hausdorff dimension and limit capacity of the nonwandering set are not continuous across the bifurcation. We also study the behaviour of equilibrium measures near the bifurcation.

A CONSTRUCTION OF THE SCRAMBLED SET WITH FULL HAUSDORFF DIMENSION FOR BETA-TRANSFORMATIONS

Fractals ◽  
2018 ◽  
Vol 26 (01) ◽  
pp. 1850005
Author(s):  
WEI-BIN LIU ◽  
CHENG HUANG ◽  
MEI-HUA LI ◽  
SHUAILING WANG
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure

For beta-transformations, we prove that the Lebesgue measure of any measurable scrambled set is zero, and there exists a scrambled set with full Hausdorff dimension.

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