Good permutations for extreme discrepancy. (English) Zbl 0768.11026

It is known that for every infinite sequence \(X: x_ 1,x_ 2,\dots\) in \([0,1)\) for the discrepancy \(D_ N\) of the first \(N\) sequence elements we have \(s(x):= \limsup_ n(D(N)/\text{Log }N) \geq 0.12\). For the usual van der Corput-sequence to base \(b\), the value \(s(x)\) tends to infinity with growing base \(b\).
In this paper it is shown that in the case of generalized van der Corput- sequences for every base \(b\) there exists a permutation \(\sigma\) such that for the generalized van der Corput-sequence \(S^ \sigma_ b\) one has \(S(S^ \sigma_ b) \leq 1/\text{Log }2\). As a special case for base \(b = 36\) a permutation \(\sigma\) is explicitly given such that \(S(S^ \sigma_{36}) = 23/(35\text{ Log }6) = 0.3667\dots\). This is the smallest value for \(s\) known till now.


11K38 Irregularities of distribution, discrepancy
Full Text: DOI


[1] Béjian, R., Minoration de la discrépance d’une suite quelconque sur T, Acta arith., 41, 2, 185-202, (1982) · Zbl 0439.10038
[2] Borel, J.P., Self similar measures and sequences, J. number theory, 31, 208-241, (1989) · Zbl 0673.10039
[3] Braaten, E.; Weller, G., An improved low discrepancy sequence for multidimensional quasi-Monte-Carlo integration, J. comput. phys., 33, 249-258, (1979) · Zbl 0426.65001
[4] Faure, H., Discrépances de suites associées à un système de numération (en dimension un), Bull. soc. math. France, 109, 143-182, (1981) · Zbl 0488.10052
[5] Kuipers, L.; Niederreiter, H., ()
[6] Lapeyre, B.; Pagès, G., Familles de suites à discrépance faible obtenues par itérations de transformations de [0, 1], C.R. acad. sci. Paris, 308, 507-509, (1989) · Zbl 0676.10038
[7] Schmidt, W.M., Irregularities of distribution VII, Acta arith., 21, 45-50, (1972) · Zbl 0244.10035
[8] Thomas, A., Discrépance en dimension un, Ann. fac. sci. Toulouse math., 10, 3, 369-399, (1989) · Zbl 0707.11054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.