<abstract xmlns="http://www.w3.org/1999/xhtml">In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the <italic>d−</italic>dimensional unit cube [Proinov, P. D.:<italic>On irregularities of distribution</italic>, C. R. Acad. Bulgare Sci. <bold>39</bold> (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.: <italic>Quantitative Theory of Uniform Distribution and Integral Approximation</italic>, 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.: <italic>On lower bounds for the</italic> ℒ2<italic>-discrepancy</italic>, J. Complexity <bold>27</bold> (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: <italic>An improved lower bound for the</italic> ℒ2<italic>-discrepancy</italic>, J. Complexity <bold>34</bold> (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. <italic>An improved lower bound for the</italic> ℒ2<italic>-discrepancy</italic>, J. Complexity <bold>34</bold> (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.</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 the ℒ2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2-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.

On Proinov’s Lower Bound for the Diaphony

Mathematical Sciences Research Centre, School of Mathematics and Physics

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"On Proinov\u2019s Lower Bound for the Diaphony"}

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...

On Proinov’s Lower Bound for the Diaphony

Published Online: Dec 25, 2020

Page range: 39 - 72

Received: Jul 20, 2020

Accepted: Sep 08, 2020

DOI: https://doi.org/10.2478/udt-2020-0010

Keywords
ℒ-discrepancy, (dyadic) diaphony, Walsh system, Haar system

© 2020 Nathan Kirk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On Proinov’s Lower Bound for the Diaphony

Published Online: Dec 25, 2020

Page range: 39 - 72

Received: Jul 20, 2020

Accepted: Sep 08, 2020

DOI: https://doi.org/10.2478/udt-2020-0010

Keywordsℒ-discrepancy, (dyadic) diaphony, Walsh system, Haar system

© 2020 Nathan Kirk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
ℒ-discrepancy, (dyadic) diaphony, Walsh system, Haar system