Dyadic diaphony of digital sequences
Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 501-521.

La diaphonie diadique est une mesure quantitative pour l’irrégularité de la distribution d’une suite dans le cube unitaire. Dans cet article nous donnons des formules pour la diaphonie diadique des (0,s)-suites digitales sur 2 , s=1,2. Ces formules montrent que, pour s{1,2} fixé, la diaphonie diadique a les mêmes valeurs pour chaque (0,s)-suite digitale. Pour s=1, il résulte que la diaphonie diadique et la diaphonie des (0,1)-suites digitales particulières sont égales, en faisant abstraction d’une constante. On détermine l’ordre asymptotique exact de la diaphonie diadique des (0,s)-suites digitales et on montre que pour s=1 elle satisfait un théorème de la limite centrale.

The dyadic diaphony is a quantitative measure for the irregularity of distribution of a sequence in the unit cube. In this paper we give formulae for the dyadic diaphony of digital (0,s)-sequences over 2 , s=1,2. These formulae show that for fixed s{1,2}, the dyadic diaphony has the same values for any digital (0,s)-sequence. For s=1, it follows that the dyadic diaphony and the diaphony of special digital (0,1)-sequences are up to a constant the same. We give the exact asymptotic order of the dyadic diaphony of digital (0,s)-sequences and show that for s=1 it satisfies a central limit theorem.

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DOI : https://doi.org/10.5802/jtnb.599
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Friedrich Pillichshammer. Dyadic diaphony of digital sequences. Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 501-521. doi : 10.5802/jtnb.599. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.599/

[1] H. Chaix and H. Faure, Discrépance et diaphonie en dimension un. Acta Arith. 63 (1993), 103–141. | MR 1206080 | Zbl 0772.11022

[2] J. Dick and F. Pillichshammer, Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces. J. Complexity 21 (2005), 149–195. | MR 2123222 | Zbl 1085.41021

[3] J. Dick and F. Pillichshammer, Dyadic diaphony of digital nets over 2 . Monatsh. Math. 145 (2005), 285–299. | MR 2162347 | Zbl pre02232176

[4] J. Dick and F. Pillichshammer, On the mean square weighted 2 -discrepancy of randomized digital (t,m,s)-nets over 2 . Acta Arith. 117 (2005), 371–403. | MR 2140164 | Zbl 1080.11058

[5] J. Dick and F. Pillichshammer, Diaphony, discrepancy, spectral test and worst-case error. Math. Comput. Simulation 70 (2005), 159–171. | MR 2176902 | Zbl pre02236013

[6] M. Drmota, G. Larcher and F. Pillichshammer, Precise distribution properties of the van der Corput sequence and related sequences. Manuscripta Math. 118 (2005), 11–41. | MR 2171290 | Zbl 1088.11060

[7] M. Drmota and R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, 1997. | MR 1470456 | Zbl 0877.11043

[8] H. Faure, Discrepancy and diaphony of digital (0,1)-sequences in prime base. Acta Arith. 117 (2004), 125–148. | MR 2139596 | Zbl 1080.11054

[9] H. Faure, Irregularites of distribution of digital (0,1)-sequences in prime base. Integers 5 (2005), A7, 12 pages. | MR 2191753 | Zbl 1084.11041

[10] V.S. Grozdanov, On the diaphony of one class of one-dimensional sequences. Internat. J. Math. Math. Sci. 19 (1996), 115–124. | MR 1361985 | Zbl 0841.11038

[11] P. Hellekalek and H. Leeb, Dyadic diaphony. Acta Arith. 80 (1997), 187–196. | MR 1450924 | Zbl 0868.11034

[12] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. John Wiley, New York, 1974. | MR 419394 | Zbl 0281.10001

[13] G. Larcher, H. Niederreiter and W.Ch. Schmid, Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatsh. Math. 121 (1996), 231–253. | MR 1383533 | Zbl 0876.11042

[14] G. Larcher and F. Pillichshammer, Sums of distances to the nearest integer and the discrepancy of digital nets. Acta Arith. 106 (2003), 379–408. | MR 1957912 | Zbl 1054.11039

[15] H. Niederreiter, Point sets and sequences with small discrepancy. Monatsh. Math. 104 (1987), 273–337. | MR 918037 | Zbl 0626.10045

[16] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. No. 63 in CBMS-NSF Series in Applied Mathematics. SIAM, Philadelphia, 1992. | MR 1172997 | Zbl 0761.65002

[17] F. Pillichshammer, Digital sequences with best possible order of L 2 –discrepancy. Mathematika 53 (2006), 149–160. | MR 2304057 | Zbl 1121.11049

[18] P.D. Proinov and V.S. Grozdanov, On the diaphony of the van der Corput-Halton sequence. J. Number Theory 30 (1988), 94–104. | MR 960236 | Zbl 0654.10050

[19] P. Zinterhof, Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden. Sitzungsber. Österr. Akad. Wiss. Math.-Natur. Kl. II 185 (1976), 121–132. | MR 501760 | Zbl 0356.65007

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