Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-06T00:25:52.950Z Has data issue: false hasContentIssue false
  • English
  • Français

LOWER BOUNDS FOR ${L}_{1} $ DISCREPANCY

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  18 March 2013

Armen Vagharshakyan
Armen Vagharshakyan*
Affiliation:
Mathematics Department, Brown University, 151 Thayer St, Providence, RI 02912, U.S.A. email [email protected]
Get access

Abstract

We find the best asymptotic lower bounds for the coefficient of the leading term of the ${L}_{1} $ norm of the two-dimensional axis-parallel discrepancy that can be obtained by Roth’s orthogonal function method among a large class of test functions. We use methods of combinatorics, probability, and complex and harmonic analysis.

MSC classification

Secondary: 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 365 - 379
Copyright
Copyright © University College London 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bilyk, D., Roth’s orthogonal function method in discrepancy theory and some new connections. In Panorama of Discrepancy Theory (eds W. W. L. Chen, A. Srivastav and G. Travaglini), Springer (2013).CrossRefGoogle Scholar
Bilyk, D., Temlyakov, V. and Yu, R., Fibonacci sets and symmetrization in discrepancy theory. J. Complexity 28 (2012), 18–36.CrossRefGoogle Scholar
Chen, W., Lectures on irregularities of point distribution (2000), http://rutherglen.ics.mq.edu.au/wchen/researchfolder/iod00.pdf.Google Scholar
Davenport, H., Note on irregularities of distribution. Mathematika 3 (1956), 131–135.CrossRefGoogle Scholar
Faure, H., Pillichshammer, F., Pirsic, G. and Schmid, W., ${L}_{2} $ discrepancy of generalized two-dimensional Hammersley point sets scrambled with arbitrary permutations. Acta Arith. 141 (2010), 395–418.CrossRefGoogle Scholar
Halasz, G., On Roth’s method in the theory of irregularities of point distributions. In Recent Progress in Analytic Number Theory, Vol. 2 (Durham, 1979), Academic Press (London, 1981), 79–94.Google Scholar
Hinrichs, A. and Markashin, L., On lower bounds for the ${L}_{2} $ discrepancy. J. Complexity 27 (2) (2011), 127–132.CrossRefGoogle Scholar
Katznelson, Y., An Introduction to Harmonic Analysis, 2nd edition, Dover (New York, 1976).Google Scholar
Macdonald, I. G., Symmetric Functions and Orthogonal Polynomials (University Lecture Series 12), American Mathematical Society (Providence, RI, 1998).Google Scholar
Matoušek, J., Geometric Discrepancy (Algorithms and Combinatorics 18), Springer (Berlin, 2010).Google Scholar
Roth, K. F., On irregularities of distribution. II. Comm. Pure Appl. Math. 29 (6) (1976), 739–744.CrossRefGoogle Scholar