×

On the diaphony of the van der Corput-Halton sequence. (English) Zbl 0654.10050

Let \(\sigma =(x_ n)^{\infty}_{n=0}\) be a set in the unit interval. Let \[ F_ N(\sigma)=(2\sum^{\infty}_{h=1}h^{-2}\quad | N^{- 1}\sum^{N-1}_{n=0}\exp (2\pi ihx_ n)|^ 2\quad)^{1/2}. \] \(F_ N(\sigma)\) is called the diaphony of \(\sigma\). The authors prove that \(F_ N(\sigma)\ll \sqrt{\log N}/N\) holds for the Van der Corput- Halton sequence. The estimation is best possible, since we know that for any set \(\sigma\), \(F_ N(\sigma)>(1/68)(\sqrt{\log N}/N)\) holds for infinitely many N.
Reviewer: Wang Yuan

MSC:

11K38 Irregularities of distribution, discrepancy
11K06 General theory of distribution modulo \(1\)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Faure, H., Discrépance quadratique de suites infinies en dimension un, (Proc. Conf. on Number Theory. Proc. Conf. on Number Theory, University of Laval, Quebec (Canada) (1987)), 22-25
[2] Haber, S., On a sequence of points of interest for numerical quadrature, J. Res. Nat. Bur. Standards (Sect. B), 70, 127-136 (1966) · Zbl 0158.16002
[3] Halton, J. H., On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math., 2, 84-90 (1960) · Zbl 0090.34505
[4] Hlawka, E., (Theorie der Gleichverteilung (1979), Bibliographisches Inst: Bibliographisches Inst Mannheim/Wien/Zurich)
[5] Kuipers, L., Simple proof of a theorem of J. F. Koksma, Nieuw Tijdschr. Wisk., 55, 108-111 (1967)
[6] Kuipers, L.; Niederreiter, H., (Uniform Distribution of Sequences (1974), Wiley: Wiley New York) · Zbl 0281.10001
[7] Niederreiter, H., Application of diophantine approximation to numerical integration, (Osgood, C. F., Diophantine Approximation and Its Applications (1973), Academic Press: Academic Press New York), 129-199
[8] Niederreiter, H., Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc., 84, 957-1041 (1978) · Zbl 0404.65003
[9] Proinov, P. D., Points of constant type and upper bound for the mean-square discrepancy of a class of infinite sequences, C. R. Acad. Bulgare Sci., 35, 753-755 (1982), [Russian] · Zbl 0494.10038
[10] Proinov, P. D., Estimation of \(L^2\) discrepancy of a class of infinite sequences, C. R. Acad. Bulgare Sci., 36, 37-40 (1983) · Zbl 0514.10039
[11] Proinov, P. D., On the \(L^2\) discrepancy of some infinite sequences, Serdica, 11, 3-12 (1985) · Zbl 0584.10033
[12] Proinov, P. D., On irregularities of distribution, C. R. Acad. Bulgare Sci., 39, No. 9, 31-34 (1986) · Zbl 0616.10042
[13] Proinov, P. D.; Grozdanov, V. S., Symmetrization of the Van der Corput-Halton sequence, C. R. Acad. Bulgare Sci., 40, No. 8, 5-8 (1987) · Zbl 0621.10035
[14] Roth, K. F., On irregularities of distribution, Mathematika, 1, 73-79 (1954) · Zbl 0057.28604
[15] Sobol’, I. M., (Multidimensional Quadrature Formulae and Haar Functions (1969), Izdat. “Nauka”: Izdat. “Nauka” Moscow), [Russian] · Zbl 0195.16903
[16] Stegbuchner, H., Zur quantitativen Theorie der Gleichverteilung mod 1, ((1980), Ber. Math. Inst. Univ. Salzburg), 9-58, Heft 3 · Zbl 0447.10037
[17] Van der Corput, J. G., Verteilungsfunktionen, (Proc. Kon. Ned. Akad. Wetensch., 38 (1935)), 813-821 · Zbl 0012.34705
[18] Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann., 77, 313-352 (1916) · JFM 46.0278.06
[19] Zinterhof, P., Über einige Abschatzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden, Sitzungsber. Österr. Akad. Wiss. Math.-Naturwiss. Kl. II, 185, 121-132 (1976) · Zbl 0356.65007
[20] Zinterhof, P.; Stegbuchner, H., Trigonometrische Approximation mit Gleichverteilungsmethoden, Studia Sci. Math. Hungar., 13, 273-289 (1978) · Zbl 0421.42001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.