The exact Hausdorff measure for random re-ordering of Cantor set (Q1895487)

Let \(X_1, X_2,\ldots\) be a sequence of i.i.d. random variables uniformly distributed on \([0,1]\), and let \(Q_n\), \(n \geq 1\), be random sets of integers defined by \(Q_n = \{m \geq 1 : X_m < X_n\}\). For a sequence of reals \(\{a_n\}^\infty_{n = 1}\) with \(\sum a_n = 1\) set \(t_n = \sum_{s \in Q_n} a_s\) and \(J_n = (t_n, t_n + a_n)\). The author finds the packing dimension for the random set \(K = [0,1] \backslash \bigcup^\infty_{n = 1} J_n\). In particular, if \(a_n = 3^{-n}\), \(n \geq 1\), then \(K\) is a random Cantor set obtained by random reordering of complementary intervals to the classical Cantor set. In this case the author derives the exact Hausdorff and packing measures for \(K\).