Pillichshammer, Friedrich Dyadic diaphony of digital sequences. (English) Zbl 1152.11036 J. Théor. Nombres Bordx. 19, No. 2, 501-521 (2007). The author considers the dyadic diaphony \(F_{2,N}(\omega)\) [see P. Hellekalek, H. Leeb, Acta Arith. 80, 2, 187–196 (1997; Zbl 0868.11034)] of digital \((0,s)\)-sequences \(\omega\) with \(s\in\{1,2\}\). For \((0,1)\)-sequences \(\omega\) he proves \[ (N\,F_{2,N}(\omega))^2=3\sum_{u=1}^\infty \left\| \frac{N}{2^u}\right\| ^2 \] and derives some interesting consequences. In particular he shows that the dyadic diaphony of \((0,1)\) sequences satisfies a central limit theorem. Similar for \((0,2)\)-sequences \(\omega\) the author obtains \[ (N\,F_{2,N}(\omega))^2=\frac 94\sum_{u=1}^\infty u\left\| \frac{N}{2^u}\right\| ^2. \] in this case the author cannot prove a central limit theorem, but a similar weaker theorem. Reviewer: Volker Ziegler (Graz) Cited in 2 ReviewsCited in 1 Document MSC: 11K06 General theory of distribution modulo \(1\) 11A63 Radix representation; digital problems 11K38 Irregularities of distribution, discrepancy 65C05 Monte Carlo methods 65C20 Probabilistic models, generic numerical methods in probability and statistics Keywords:discrepancy of sequences; dyadic diaphony Citations:Zbl 0868.11034 PDFBibTeX XMLCite \textit{F. Pillichshammer}, J. Théor. Nombres Bordx. 19, No. 2, 501--521 (2007; Zbl 1152.11036) Full Text: DOI Numdam Numdam EuDML References: [1] H. Chaix and H. Faure, Discrépance et diaphonie en dimension un. Acta Arith. 63 (1993), 103-141. · 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. · Zbl 1085.41021 [3] J. Dick and F. Pillichshammer, Dyadic diaphony of digital nets over \({\mathbb{Z}}_2\). Monatsh. Math. 145 (2005), 285-299. · Zbl 1130.11042 [4] J. Dick and F. Pillichshammer, On the mean square weighted \(\mathcal{L}_2\)-discrepancy of randomized digital \((t,m,s)\)-nets over \({\mathbb{Z}}_2\). Acta Arith. 117 (2005), 371-403. · Zbl 1080.11058 [5] J. Dick and F. Pillichshammer, Diaphony, discrepancy, spectral test and worst-case error. Math. Comput. Simulation 70 (2005), 159-171. · Zbl 1193.65003 [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. · Zbl 1088.11060 [7] M. Drmota and R.F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, 1997. · Zbl 0877.11043 [8] H. Faure, Discrepancy and diaphony of digital \((0,1)\)-sequences in prime base. Acta Arith. 117 (2004), 125-148. · Zbl 1080.11054 [9] H. Faure, Irregularites of distribution of digital \((0,1)\)-sequences in prime base. Integers 5 (2005), A7, 12 pages. · 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. · Zbl 0841.11038 [11] P. Hellekalek and H. Leeb, Dyadic diaphony. Acta Arith. 80 (1997), 187-196. · Zbl 0868.11034 [12] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. John Wiley, New York, 1974. · 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. · 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. · Zbl 1054.11039 [15] H. Niederreiter, Point sets and sequences with small discrepancy. Monatsh. Math. 104 (1987), 273-337. · 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. · Zbl 0761.65002 [17] F. Pillichshammer, Digital sequences with best possible order of \(L_2\)-discrepancy. Mathematika 53 (2006), 149-160. · 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. · 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. · Zbl 0356.65007 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.