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

@article{AIHPB_2014__50_4_1371_0,
     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},
     volume = {50},
     number = {4},
     year = {2014},
     doi = {10.1214/13-AIHP556},
     mrnumber = {3269998},
     zbl = {06377558},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP556/}
}

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
SP  - 1371
EP  - 1384
VL  - 50
IS  - 4
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP556/
DO  - 10.1214/13-AIHP556
LA  - en
ID  - AIHPB_2014__50_4_1371_0
ER  -

%0 Journal Article
%A Järvenpää, Esa
%A Järvenpää, Maarit
%A Koivusalo, Henna
%A Li, Bing
%A Suomala, Ville
%T Hausdorff dimension of affine random covering sets in torus
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 1371-1384
%V 50
%N 4
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP556/
%R 10.1214/13-AIHP556
%G en
%F AIHPB_2014__50_4_1371_0

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://archive.numdam.org/articles/10.1214/13-AIHP556/

Bibliographie
Cité par

[1] V. Beresnevich and S. Velani. A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164 (2006) 971-992. | MR | Zbl

[2] J. Barral and A.-H. Fan. Covering numbers of different points in Dvoretzky covering. Bull. Sci. Math. 129 (4) (2005) 275-317. | MR | Zbl

[3] P. Billard. Séries de Fourier aléatoirement bornées, continues, uniformément convergentes. Ann. Sci. École Norm. Sup. (3) 82 (1965) 131-179. | Numdam | MR | Zbl

[4] A. Durand. On randomly placed arcs on the circle. In Recent Developments in Fractals and Related Fields 343-351. Appl. Numer. Harmon. Anal. Birkhäuser, Boston, 2010. | MR | Zbl

[5] A. Dvoretzky. On covering a circle by randomly placed arcs. Proc. Natl. Acad. Sci. USA 42 (1956) 199-203. | MR | Zbl

[6] Y. El Hélou. Recouvrement du tore $T^{q}$ par des ouverts aléatoires et dimension de Hausdorff de l’ensemble non recouvert. C. R. Acad. Sci. Paris Sér. A-B 287 (1978) A815-A818. | MR | Zbl

[7] P. Erdős. Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961) 221-254. | MR | Zbl

[8] K. J. Falconer. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103 (1988) 339-350. | MR | Zbl

[9] A.-H. Fan. How many intervals cover a point in Dvoretzky covering? Israel J. Math. 131 (2002) 157-184. | MR | Zbl

[10] A.-H. Fan and J.-P. Kahane. Rareté des intervalles recouvrant un point dans un recouvrement aléatoire. Ann. Inst. Henri Poincaré Probab. Stat. 29 (1993) 453-466. | Numdam | MR | Zbl

[11] A.-H. Fan, J. Schmeling and S. Troubetzkoy. A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation. Proc. London Math. Soc. (3) 107 (2013) 1173-1219. | MR

[12] A.-H. Fan and J. Wu. On the covering by small random intervals. Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 125-131. | Numdam | MR | Zbl

[13] J. Hawkes. On the covering of small sets by random intervals. Quart. J. Math. Oxford Ser. (2) 24 (1973) 427-432. | MR | Zbl

[14] J. Hawkes. On the asymptotic behaviour of sample spacings. Math. Proc. Cambridge Philos. Soc. 90 (2) (1985) 293-303. | MR | Zbl

[15] J. Hoffmann-Jørgensen. Coverings of metric spaces with randomly placed balls. Math. Scand. 32 (1973) 169-186. | MR | Zbl

[16] S. Janson. Random coverings in several dimensions. Acta Math. 156 (1986) 83-118. | MR | Zbl

[17] J. Jonasson and J. E. Steif. Dynamical models for circle covering: Brownian motion and Poisson updating. Ann. Probab. 36 (2008) 739-764. | MR | Zbl

[18] J.-P. Kahane. Sur le recouvrement d'un cercle par des arcs disposés au hasard. C. R. Acad. Sci. Paris 248 (1956) 184-186. | MR | Zbl

[19] J.-P. Kahane. Some Random Series of Functions. Cambridge Studies in Advanced Mathematics 5. Cambridge Univ. Press, Cambridge, 1985. | MR | Zbl

[20] J.-P. Kahane. Recouvrements aléatoires et théorie du potentiel. Colloq. Math. 60/61 (1990) 387-411. | MR | Zbl

[21] J.-P. Kahane. Random coverings and multiplicative processes. In Fractal Geometry and Stochastics II 125-146. Progr. Probab. 46. Birkhäuser, Basel, 2000. | MR | Zbl

[22] B. Li, N.-R. Shieh and Y.-M. Xiao. Hitting probabilities of the random covering sets. In Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. II. Fractals in Applied Mathematics 307-323. Contemp. Math. 601. Amer. Math. Soc., Providence, RI, 2013. | MR

[23] L. Liao and S. Seuret. Diophantine approximation by orbits of Markov maps. Ergodic Theory Dynam. Systems 33 (2013) 585-608. | MR | Zbl

[24] B. Mandelbrot. On Dvoretzky coverings for the circle. Z. Wahrsch. Verw. Gebiete 22 (1972) 158-160. | MR | Zbl

[25] B. Mandelbrot. Renewal sets and random cutouts. Z. Wahrsch. Verw. Gebiete 22 (1972) 145-157. | MR | Zbl

[26] P. Mattila. Geometry of Sets and Measures in Euclidean Spaces. Cambridge Univ. Press, Cambridge, 1995. | MR | Zbl

[27] T. Orponen. On the packing dimension and category of exceptional sets of orthogonal projections. Available at http://arxiv.org/abs/1204.2121v3.

[28] L. A. Shepp. Covering the line with random intervals. Z. Wahrsch. Verw. Gebiete 23 (1972) 163-170. | MR | Zbl

[29] L. A. Shepp. Covering the circle with random arcs. Israel J. Math. 11 (1972) 328-345. | MR | Zbl

Cité par Sources :