On the covering by small random intervals. (Q1426661)

Let \(a>0\) and \(\alpha >1\), and for each \(n\geq 0\), let \(\ell_n=a/n^{\alpha}\). If \((w_n)_{n\geq 0}\) is a sequence of independent random variables uniformly distributed over the unit interval \([0,1)\) and if for each \(n\), we set \(I_n=w_n+(0,\ell_n)\) (modulo 1), then \((I_n)_{n\geq 0}\) is a sequence of random intervals. The authors consider the set \(E_{\infty}(w)=\limsup I_n\) of points in the unit interval which are covered infinitely often, and they show that the Hausdorff dimension of this set is almost surely equal to \(1/\alpha\).