Comparison Between LS-Sequences and $\beta$-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.