Comparison Between LS-Sequences and $$\beta $$ β -Adic van der Corput Sequences

Abstract

In 2011 the author introduced a generalization of van der Corput sequences, the so called LS-sequences defined for integers L, S such that \(L \ge 1\), \(S\ge 0\), \(L+ S \ge 2\), and \(\gamma \in ]0,1[\) is the positive solution of \(S\gamma ^2 + L\gamma =1\). These sequences coincide with the classical van der Corput sequences whenever \(S=0\), are uniformly distributed for all L, S and have low discrepancy when \(L\ge S\). In this paper we compare the LS-sequences and the \(\beta \)-adic van der Corput sequences where \(\beta >1\) is the Pisot root of \(x^2-Lx-L\). Using a suitable numeration system \(G=\{ G_{n}\}_{n \ge 0}\), where the base sequence is the linear recurrence of order two, \(G_{n+2}=L G_{n+1}+LG_n\), with initial conditions \(G_0=1\) and \(G_1=L+1\), we prove that when \(L=S\) the (L, L)-sequence with \(L\gamma ^2 + L\gamma =1\) and the \(\beta \)-adic van der Corput sequence with \(\beta =1/{\gamma }\) and \(\beta ^2=L\beta +L\) can be obtained from each other by a permutation. In particular for \(\beta =\varPhi \), the golden ratio, the \(\beta \)-adic van der Corput sequence coincides with the Kakutani–Fibonacci sequence obtained for \(L=S=1\), which has been already studied.