I would like to present a joint work, with Radhakrishnan Nair, on the characterization of unique ergodicity on some subsequences of the natural numbers called Hartmann uniformly distributed sequences. Then we shall see an application of the characterization theorem on answering a question of O. Strauch, but in a more general framework, regarding the limit distribution of consecutive elements of the van der Corput sequence. Recently, C. Aistleitner and M. Hofer calculated the asymptotic distribution function of $(\Phi_b(n), \Phi_b(n+1), \dots, \Phi_b(n+s-1))_{n=0}^\infty$ on $[0,1)^s$, where $(\Phi_b(n))_{n=0}^\infty$ denotes the van der Corput sequence in base $b>1$, and showed that it is a copula. In the talk, we shall see that this phenomenon extends not only to a broad class of subsequences of the van der Corput sequence but also to a more general setting in the a-adic numbers. Indeed, we shall use the characterization of unique ergodicity, together with the fact that the van der Corput sequence can be seen as the orbit of the origin under the ergodic Kakutani-von Neumann transformation.