On the \(n\)-point correlation of van der Corput sequences (Q6564338)

The \(N\)-point correlation \(F_N(s)\) of a sequence \((x_n)_{n\ge1}\) of elements in \([0,1]\) counts the number of distinct pairs \(j,k \le N\) such that \(\mathrm{dist}(x_k-x_j,\mathbb{Z}) \le s/N\), divided by \(N\). For the van der Corput sequence in base 2, the author gives the explicit formula\N\[\NF_N(s) = \frac{1}{N} \sum_{j=0}^M \bigg(\bigg\lfloor\frac{s}{N} 2^j\bigg\rfloor + \sum_{k=j+1}^M 2e_k \bigg\lceil\frac{1}{2} \bigg\lfloor\frac{s}{N} 2^{k+1}\bigg\rfloor\bigg\rceil\bigg) e_j 2^{j+1},\N\]\Nwhere \(N = \sum_{j=0}^M e_j 2^j\), from which he derives that \(\lim_{n\to\infty} F_N(s)\) exists if and only if \(0 \le s \le 1/2\), with the limit being 0 in this case.