On the covering by small random intervals
نویسندگان
چکیده
Consider the random intervals In = ωn+ (0, n) (modulo 1) with their left points ωn independently and uniformly distributed over the interval [0,1)=R/Z and with their lengths decreasing to zero. We prove that the Hausdorff dimension of the set limnIn of points covered infinitely often is almost surely equal to 1/α when n = a/nα for some a > 0 and α > 1. 2003 Elsevier SAS. All rights reserved. Résumé Considérons des intervalles aléatoires In = ωn + (0, n) (modulo 1) dont les extrémités gauches ωn sont indépendantes et uniformément réparties sur l’intervalle [0,1) = R/Z et dont les longueurs décroissent vers zéro. Nous montrons que la dimension de Hausdorff de l’ensemble limnIn des points infiniment recouverts est presque sûrement égale à 1/α quand n = a/nα avec a > 0 et α > 1. 2003 Elsevier SAS. All rights reserved. MSC: 60D05; 52C17; 28A80
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