On some remarkable properties of the two-dimensional Hammersley point set in base 2

Peter Kritzer[1]

  • [1] Department of Mathematics University of Salzburg Hellbrunnerstr. 34 A-5020 Salzburg, Austria

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 1, page 203-221
  • ISSN: 1246-7405

Abstract

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We study a special class of ( 0 , m , 2 ) -nets in base 2. In particular, we are concerned with the two-dimensional Hammersley net that plays a special role among these since we prove that it is the worst distributed with respect to the star discrepancy. By showing this, we also improve an existing upper bound for the star discrepancy of digital ( 0 , m , 2 ) -nets over ℤ 2 . Moreover, we show that nets with very low star discrepancy can be obtained by transforming the Hammersley point set in a suitable way.

How to cite

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Kritzer, Peter. "On some remarkable properties of the two-dimensional Hammersley point set in base 2." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 203-221. <http://eudml.org/doc/249629>.

@article{Kritzer2006,
abstract = {We study a special class of $(0,m,2)$-nets in base 2. In particular, we are concerned with the two-dimensional Hammersley net that plays a special role among these since we prove that it is the worst distributed with respect to the star discrepancy. By showing this, we also improve an existing upper bound for the star discrepancy of digital $(0,m,2)$-nets over $\mathbb\{Z\}_\{2\}$. Moreover, we show that nets with very low star discrepancy can be obtained by transforming the Hammersley point set in a suitable way.},
affiliation = {Department of Mathematics University of Salzburg Hellbrunnerstr. 34 A-5020 Salzburg, Austria},
author = {Kritzer, Peter},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Hammersley point set; star discrepancy; digital ; m; 2); digital shifts},
language = {eng},
number = {1},
pages = {203-221},
publisher = {Université Bordeaux 1},
title = {On some remarkable properties of the two-dimensional Hammersley point set in base 2},
url = {http://eudml.org/doc/249629},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Kritzer, Peter
TI - On some remarkable properties of the two-dimensional Hammersley point set in base 2
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 203
EP - 221
AB - We study a special class of $(0,m,2)$-nets in base 2. In particular, we are concerned with the two-dimensional Hammersley net that plays a special role among these since we prove that it is the worst distributed with respect to the star discrepancy. By showing this, we also improve an existing upper bound for the star discrepancy of digital $(0,m,2)$-nets over $\mathbb{Z}_{2}$. Moreover, we show that nets with very low star discrepancy can be obtained by transforming the Hammersley point set in a suitable way.
LA - eng
KW - Hammersley point set; star discrepancy; digital ; m; 2); digital shifts
UR - http://eudml.org/doc/249629
ER -

References

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  1. L. De Clerck, A method for exact calculation of the stardiscrepancy of plane sets applied to the sequences of Hammersley. Monatsh. Math. 101 (1986), 261–278. Zbl0588.10059MR851948
  2. J. Dick, P. Kritzer, Star-discrepancy estimates for digital ( t , m , 2 ) -nets and ( t , 2 ) -sequences over ℤ 2 . Acta Math. Hungar. 109 (3) (2005), 239–254. Zbl1102.11036MR2187287
  3. M. Drmota, R. F. Tichy, Sequences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer, Berlin, 1997. Zbl0877.11043MR1470456
  4. H. Faure, On the star-discrepancy of generalized Hammersley sequences in two dimensions. Monatsh. Math. 101 (1986), 291–300. Zbl0588.10060MR851950
  5. J. H. Halton, S. K. Zaremba, The extreme and the L 2 discrepancies of some plane sets. Monatsh. Math. 73 (1969), 316–328. Zbl0183.31401MR252329
  6. L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences. John Wiley, New York, 1974. Zbl0281.10001MR419394
  7. G. Larcher, F. Pillichshammer, Sums of distances to the nearest integer and the discrepancy of digital nets. Acta Arith. 106 (2003), 379–408. Zbl1054.11039MR1957912
  8. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. CBMS–NSF Series in Applied Mathematics 63, SIAM, Philadelphia, 1992. Zbl0761.65002MR1172997
  9. F. Zhang, Matrix Theory. Springer, New York, 1999. Zbl0948.15001MR1691203

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