Weak multipliers for generalized van der Corput sequences

Generalized van der Corput sequences are onedimensional, infinite sequences in the unit interval. They are generated from permutations in integer base b and are the building blocks of the multi-dimensional Halton sequences. Motivated by recent progress of Atanassov on the uniform distribution behavior of Halton sequences, we study, among others, permutations of the form P (i) = ai (mod b) for coprime integers a and b. We show that multipliers a that either divide b − 1 or b + 1 generate van der Corput sequences with weak distribution properties. We give explicit lower bounds for the asymptotic distribution behavior of these sequences and relate them to sequences generated from the identity permutation in smaller bases, which are, due to Faure, the weakest distributed generalized van der Corput sequences. Manuscrit reçu le 12 février 2012, révisé le 25 avril 2012. This work is supported by the Graduate School of IST Austria (Institute of Science and Technology Austria). Mots clefs. Uniform distribution, diaphony, generalized van der Corput sequence. Classification math. 11K06, 11K38. 730 Florian Pausinger