1. bookVolume 15 (2020): Issue 2 (December 2020)
Journal Details
License
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English
access type Open Access

On Proinov’s Lower Bound for the Diaphony

Published Online: 25 Dec 2020
Volume & Issue: Volume 15 (2020) - Issue 2 (December 2020)
Page range: 39 - 72
Received: 20 Jul 2020
Accepted: 08 Sep 2020
Journal Details
License
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English
Abstract

In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the2-discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

Keywords

MSC 2010

[1] BILYK, D.: On Roth’s orthogonal function method in discrepancy theory, Unif. Distrib. Theory 6 (2011), no. 1, 143–184.Search in Google Scholar

[2] CRISTEA, L. L.—PILLICHSHAMMER, F.: A lower bound for the b−adic diaphony, Rend. Mat. Appl. Ser. VII 27 (2007), 147-153.Search in Google Scholar

[3] DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511761188Search in Google Scholar

[4] DICK, J.—PILLICHSHAMMER, F.: Explicit constructions of point sets and sequences with low discrepancy, (P. Kritzer, ed. et al.), In: Uniform Distribution and Quasi-Monte Carlo Methods. Discrepancy, Integration and Applications. Radon Ser. Comput. Appl. Math. Vol. 15, De Gruyter, Berlin 2014, pp. 63–86.10.1515/9783110317930.63Search in Google Scholar

[5] ERDŐS, P.—TURÁN, P.: On a problem in the theory of uniform distribution, Indag. Math. 10 (1948), 370–378.Search in Google Scholar

[6] FINE, N. J.: On the Walsh Functions, Trans. Amer. Math. Soc. 65, No. 3 (1949), 372–414.Search in Google Scholar

[7] GROZDANOV, V.—STOILOVA, S.: On the theory of b-adic diaphony, C. R. Acad. Bulgare Sci. 54 (2001), 31–34.Search in Google Scholar

[8] HELLEKALEK, P.—LEEB, H.: Dyadic diaphony, Acta. Arith. 80 (1997), no. 2, 187–196.Search in Google Scholar

[9] HINRICHS, A.—LARCHER, G.: An improved lower bound for the2-discrepancy, J. Complexity 34 (2016), 68–77.10.1016/j.jco.2015.11.008Search in Google Scholar

[10] HINRICHS, A.—MARKHASIN, L.: On lower bounds for the2-discrepancy, J. Complexity 27 (2011), 127–132.10.1016/j.jco.2010.11.002Search in Google Scholar

[11] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. In: Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.Search in Google Scholar

[12] LARCHER, G.: On the star-discrepancy of sequences in the unit-interval, J. Complexity 31 (2015), no. 3, 474–485.Search in Google Scholar

[13] LARCHER, G.: On the discrepancy of sequences in the unit-interval, Indag. Math. (N.S.) 27 (2016), 546–558.10.1016/j.indag.2015.11.003Search in Google Scholar

[14] LARCHER, G.: Digital Point Sets: Analysis and Application. In: Random and Quasi-Random Point Sets, Lect. Notes in Stat. Vol. 138, Springer-Verlag, Berlin, 1998, pp. 167–222.Search in Google Scholar

[15] LARCHER, G.—PUCHHAMMER, F.: An improved bound for the star discrepancy of sequences in the unit interval, Unif. Distrib. Theory 11 (2016), no. 1, 1–14.Search in Google Scholar

[16] PAUSINGER, F.: On the intriguing search for good permutations, Unif. Distrib. Theory 14 (2019), no. 1, 53–86.Search in Google Scholar

[17] PROINOV, P. D.: On irregularities of distribution, C.R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34.Search in Google Scholar

[18] PROINOV, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000). (In Bulgarian)Search in Google Scholar

[19] PROINOV, P. D.: On extreme and2-discrepancies of symmetric finite sequences, Serdica Math. J. 10 (1984), 376–383.Search in Google Scholar

[20] PROINOV, P. D.: On the2-discrepancy of some infinite sequences, Serdica Math. J. 11, (1985), 3–12.Search in Google Scholar

[21] ROTH, K. F.: On irregularities of distribution, Mathematika 1 (1954), 73–79.10.1112/S0025579300000541Search in Google Scholar

[22] VAN DER CORPUT, J. G.: Verteilungsfunktionen I, Proc. Akad. Amsterdam 38 (1935), 813–821. (In German)Search in Google Scholar

[23] VAN DER CORPUT, J. G.: Verteilungsfunktionen II, Proc. Akad. Amsterdam 38 (1935), 1058–1066. (In German)Search in Google Scholar

[24] WALSH, J. L.: A closed set of normal orthogonal functions, Amer. J. Math. 55 (1923), 5–24.10.2307/2387224Search in Google Scholar

[25] WEYL, H.: Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352. (In German)Search in Google Scholar

[26] ZINTERHOF, P.:Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden, Sitzungsber. Österr. Akad. Wiss. Math.-Naturwiss. Kl. S.-B. II 185 (1976), 121–132. (In German)Search in Google Scholar

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