The exact Hausdorff measure for random re-ordering of Cantor set
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 \(K\).
scientific article; zbMATH DE number 783472