Hitting probabilities of random covering sets in tori and metric spaces

Järvenpää, Esa; Järvenpää, Maarit; Koivusalo, Henna; Li, Bing; Суомала, Вилле; Xiao, Yimin

doi:10.1214/16-ejp4658

Cited by 17 publications

(15 citation statements)

References 51 publications

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“…For the hitting probability of these random Cantor sets, we consider the special case that M k = M and N k = N for all k ∈ N, and we have the following result. Note that the result is similar to the hitting probability of fractal percolation (see [28,Theorem 9.5 ]) and random covering sets (see [16]). (2) If α > d − s then E intersects F with positive probability.…”

Section: 4supporting

confidence: 68%

A Class of Random Cantor Sets

Chen¹

2017

Real Analysis Exchange

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In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category. For the natural random measures on these random Cantor sets, we consider their almost sure lower and upper local dimensions. In the end we study the hitting probabilities of a special subclass of these random Cantor sets.2010 Mathematics Subject Classification. 28A80, 37F40, 60D05.

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Section: 4supporting

confidence: 68%

A Class of Random Cantor Sets

Chen¹

2017

Real Analysis Exchange

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“…The Dvoretzky problem has recently regained some attention with numerous interesting generalisations. For instance, refinements on the covering frequencies have been obtained by Barral and Fan [5], the probability of hitting a given analytic set has been studied in [20], coverings with arbitrary sets with non-empty interiors (not necessarily balls) in Ahlfors regular metric spaces are considered in [16,17,23], and finally in [14] the authors focus on coverings of smooth Riemann manifolds M by measurable sets distributed according to probability measures not purely singular with respect to the natural measure on M (see also the references in the above mentioned papers). A related problem concerns the random cut-out sets with inhomogeneous densities in [21].…”

Section: Introductionmentioning

confidence: 99%

Inhomogeneous random coverings of topological Markov shifts

Seuret¹

2017

Math. Proc. Camb. Phil. Soc.

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Let $\mathscr{S}$ be an irreducible topological Markov shift, and let μ be a shift-invariant Gibbs measure on $\mathscr{S}$. Let (Xn)n ≥ 1 be a sequence of i.i.d. random variables with common law μ. In this paper, we focus on the size of the covering of $\mathscr{S}$ by the balls B(Xn, n−s). This generalises the original Dvoretzky problem by considering random coverings of fractal sets by non-homogeneously distributed balls. We compute the almost sure dimension of lim supn →+∞B(Xn, n−s) for every s ≥ 0, which depends on s and the multifractal features of μ. Our results include the inhomogeneous covering of $\mathbb{T}^d$ and Sierpinski carpets.

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“…That being said, in the Euclidean setting a comprehensive description of the P-almost sure metric properties of Λ((φ j ), Υ) is given in [11]. See also [14] and [19]. Our application below holds in the more general metric space setting and also provides an alternative proof for some of the important results appearing in [11].…”

Section: An Application Of Theorem 1: Random Lim Sup Setsmentioning

confidence: 96%

A general mass transference principle

Allen

Baker

2019

Sel. Math. New Ser.

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The Mass Transference Principle proved by Beresnevich and Velani in 2006 is a celebrated and highly influential result which allows us to infer Hausdorff measure statements for lim sup sets of balls in R n from a priori weaker Lebesgue measure statements. The Mass Transference Principle and subsequent generalisations have had a profound impact on several areas of mathematics, especially Diophantine Approximation. In the present paper, we prove a considerably more general form of the Mass Transference Principle which extends known results of this type in several distinct directions. In particular, we establish a mass transference principle for lim sup sets defined via neighbourhoods of sets satisfying a certain local scaling property. Such sets include self-similar sets satisfying the open set condition and smooth compact manifolds embedded in R n . Furthermore, our main result is applicable in locally compact metric spaces and allows one to transfer Hausdorff g-measure statements to Hausdorff f -measure statements. We conclude the paper with an application of our mass transference principle to a general class of random lim sup sets.2000 Mathematics Subject Classification: Primary 11J83, 28A78; Secondary 11K60.

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Hitting probabilities of random covering sets in tori and metric spaces

Cited by 17 publications

References 51 publications

A Class of Random Cantor Sets

A Class of Random Cantor Sets

Inhomogeneous random coverings of topological Markov shifts

A general mass transference principle

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