Unique ergodicity and distribution functions for subsequences of the van der Corput sequence

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 regarding the limit distribution of consecutive elements of the van der Corput sequence. Recently, C. Aistleitner and M. Hofer calculted the asymptotic distribution function of $(\Phi_b(n), \Phi_b(n+1), \dotsc, \Phi_b(n+s-1))_{n=0}^\infty$ 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 to a broad class of subsequences of the van der Corput sequence. 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.