Hausdorff dimension of affine random covering sets in torus

Järvenpää, Esa; Järvenpää, Maarit; Koivusalo, Henna; Li, Bing; Suomala, Ville

doi:10.1214/13-AIHP556

Järvenpää, Esa ; Järvenpää, Maarit ; Koivusalo, Henna ; Li, Bing ; Suomala, Ville

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1371-1384.

Résumé
Abstract

Nous calculons presque sûrement la dimension de Hausdorff de l’ensemble de recouvrement aléatoire ${lim sup}_{n \to \infty} (g_{n} + ξ_{n})$ dans le tore $𝕋^{d}$ de dimension $d$ , où $g_{n} \subset 𝕋^{d}$ sont des parallélépipèdes, ou plus généralement, des images linéaires d’un ensemble d’intérieur non vide et $ξ_{n} \in 𝕋^{d}$ sont des points aléatoires indépendants et uniformément distribués. La formule de dimension, exprimée en fonction des valeurs singulières des applications linéaires, est valable à condition que la suite de ces valeurs singulières soit décroissante.

We calculate the almost sure Hausdorff dimension of the random covering set ${lim sup}_{n \to \infty} (g_{n} + ξ_{n})$ in $d$ -dimensional torus $𝕋^{d}$ , where the sets $g_{n} \subset 𝕋^{d}$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $ξ_{n} \in 𝕋^{d}$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

MR Zbl | 1 citation dans Numdam

DOI : 10.1214/13-AIHP556

Classification : 60D05, 28A80
Mots-clés : random covering set, Hausdorff dimension, affine Cantor set

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     author = {J\"arvenp\"a\"a, Esa and J\"arvenp\"a\"a, Maarit and Koivusalo, Henna and Li, Bing and Suomala, Ville},
     title = {Hausdorff dimension of affine random covering sets in torus},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1371--1384},
     publisher = {Gauthier-Villars},
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     number = {4},
     year = {2014},
     doi = {10.1214/13-AIHP556},
     mrnumber = {3269998},
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     url = {http://www.numdam.org/articles/10.1214/13-AIHP556/}
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Järvenpää, Esa; Järvenpää, Maarit; Koivusalo, Henna; Li, Bing; Suomala, Ville. Hausdorff dimension of affine random covering sets in torus. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1371-1384. doi : 10.1214/13-AIHP556. http://www.numdam.org/articles/10.1214/13-AIHP556/

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