Discrepancy bounds for nonoverlapping pairs of quadratic congruential pseudorandom numbers (Q1901980)

In this paper the quadratic congruential generator is investigated: \[ y_{n+1} \equiv ay^2_n+ by_n+ c\pmod m, \qquad n\geq 0, \] where \(a\), \(b\), \(c\), \(y_0\) are given parameters in \(\mathbb{Z}_m\). Setting \(x_n= {1\over m} y_n\) one produces a quasi-random sequence in \([0,1)\), provided that the parameters are suitably chosen. The author considers pairs \({\mathbf x}_n= (x_{2n}, x_{2n+1})\) of subsequent elements and proves bounds for the two-dimensional discrepancy \(D_{m/2} ({\mathbf x}_n)\). This continues earlier investigations of H. Niederreiter and the author.