Hausdorff dimension of affine random covering sets in torus

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.

Nous calculons presque sûrement la dimension de Hausdorff de l’ensemble de recouvrement aléatoire ${lim\; sup}_{n\to \infty }(g_n+{\xi }_n)$ 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 ${\xi }_n\in {𝕋}^d$ 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 ${lim\; sup}_{n\to \infty }(g_n+{\xi }_n)$ 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 ${\xi }_n\in {𝕋}^d$ 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.

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