On the limit distribution of consecutive elements of the van der Corput sequence

Recently, Fialov´a and Strauch, Uniform Distribution Theory, 6(1):101-125, 2011, calculated the asymptotic distribution function (adf) of the two-dimensional sequence (φb(n), φb(n + 1))n≥0, where (φb(n))n≥0 denotes the van der Corput sequence in base b. In the present paper we solve the general problem asking for the limit distribution of (φb(n), φb(n+1), . . . , φb(n+s−1))n≥0.
We use the fact that the van der Corput sequence can be seen as the orbit of the
origin under the ergodic von Neumann-Kakutani transformation.