Hausdorff dimension of affine random covering sets in torus

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

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1371-1384
  • ISSN: 0246-0203

Abstract

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We calculate the almost sure Hausdorff dimension of the random covering set lim sup n → ∞ ( g n + ξ n ) in d -dimensional torus 𝕋 d , where the sets g n ⊂ 𝕋 d are parallelepipeds, or more generally, linear images of a set with nonempty interior, and ξ n ∈ 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.

How to cite

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Järvenpää, Esa, et al. "Hausdorff dimension of affine random covering sets in torus." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1371-1384. <http://eudml.org/doc/272045>.

@article{Järvenpää2014,
abstract = {We calculate the almost sure Hausdorff dimension of the random covering set $\limsup _\{n\rightarrow \infty \}(g_\{n\}+\xi _\{n\})$ in $d$-dimensional torus $\mathbb \{T\}^\{d\}$, where the sets $g_\{n\}\subset \mathbb \{T\}^\{d\}$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $\xi _\{n\}\in \mathbb \{T\}^\{d\}$ 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.},
author = {Järvenpää, Esa, Järvenpää, Maarit, Koivusalo, Henna, Li, Bing, Suomala, Ville},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random covering set; Hausdorff dimension; affine Cantor set},
language = {eng},
number = {4},
pages = {1371-1384},
publisher = {Gauthier-Villars},
title = {Hausdorff dimension of affine random covering sets in torus},
url = {http://eudml.org/doc/272045},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Järvenpää, Esa
AU - Järvenpää, Maarit
AU - Koivusalo, Henna
AU - Li, Bing
AU - Suomala, Ville
TI - Hausdorff dimension of affine random covering sets in torus
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1371
EP - 1384
AB - We calculate the almost sure Hausdorff dimension of the random covering set $\limsup _{n\rightarrow \infty }(g_{n}+\xi _{n})$ in $d$-dimensional torus $\mathbb {T}^{d}$, where the sets $g_{n}\subset \mathbb {T}^{d}$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $\xi _{n}\in \mathbb {T}^{d}$ 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.
LA - eng
KW - random covering set; Hausdorff dimension; affine Cantor set
UR - http://eudml.org/doc/272045
ER -

References

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