On the covering by small random intervals

Consider the random intervals I-n = omega(n) + (0, ln) (modulo 1) with their left points omega(n) 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 (lim) over bar I-n(n) of points covered infinitely often is almost surely equal to 1/alpha when l(n) = a/n(alpha) for some a > 0 and alpha > 1. (C) 2003 Elsevier SAS. All rights reserved.