Have a personal or library account? Click to login

On the Discrepancy of Two Families of Permuted Van der Corput Sequences

Open Access
|Jul 2018

References

  1. [1] BOURGAIN, J.—KONTOROVICH, A.: On Zaremba’s conjecture, Ann. of Math. 180 (2014), no. 2. 1–6
  2. [2] CARLITZ, L.: Permutations in a finite field, Proc. Amer. Math. Soc. 4 (1953), 538.10.1090/S0002-9939-1953-0055965-8
  3. [3] CHAIX, H.—FAURE, H.: Discrépance et diaphonie en dimension un, Acta Arith. 63 (1993), 103–141.10.4064/aa-63-2-103-141
  4. [4] DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences. Cambridge Univ. Press, Cambridge, England, 2010.10.1017/CBO9780511761188
  5. [5] DRMOTA, M.—TICHY, R. F.: Sequences, Discrepancies and Applications. In: Lecture Notes in Math. Vol. 1651. Springer-Verlag, Berlin, 1997.
  6. [6] FAURE, H.: Discrépance de suites associéesàunsystème de numération (en dimension un), Bull. Soc. Math. France 109 (1981), no 2, 143–182.
  7. [7] FAURE, H.: Good permutations for extreme discrepancy, J. Number Theory 42 (1992), 47–56.10.1016/0022-314X(92)90107-Z
  8. [8] FAURE, H.: Irregularities of distribution of digital (0, 1)-sequences in prime base, Integers 5 (2005), no. 3, A7, 12 pages.
  9. [9] FAURE, H.: Selection criteria for (random) generation of digital (0,s)-sequences. In: Monte Carlo and Quasi-Monte Carlo Methods 2004, (H. Niederreiter and D. Talay, eds.), Springer-Verlag, Berlin (2006), pp. 113–126.
  10. [10] FAURE, H.: Star extreme discrepancy of generalized two-dimensional Hammersley point sets, Unif. Distrib. Theory 3 (2008), no. 2, 45–65.
  11. [11] FAURE, H—KRITZER, P.—PILLICHSHAMMER, F.: From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules, Indag. Math. 26 (2015), 760–822.10.1016/j.indag.2015.09.001
  12. [12] FAURE, H.—LEMIEUX, C.: Generalized Halton Sequences in 2008: A Comparative Study, ACM Trans. Model. Comp. Sim. 19 (2009), no. 15, 1–31.
  13. [13] HUANG, S.: An Improvement to Zaremba’s Conjecture. Geometric and Functional Analysis 25 (2015), 860–914.
  14. [14] KHINCHIN, A. YA.: Continued Fractions. The University of Chicago Press, Chicago, Ill.-London, 1964.10.1063/1.3051235
  15. [15] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.
  16. [16] LARCHER, G.: On the discrepancy of sequences in the unit-interval, Indag. Math., New Series 27 (2016), 546–558.
  17. [17] MATOUŠEK, J.: On the L2-discrepancy for anchored boxes, J. Complexity 14 (1998), 527–556.10.1006/jcom.1998.0489
  18. [18] NIEDERREITER, H.: Applications of diophantine approximations to numerical integration, In: Diophantine Approximation and Its Applications, (C.F. Osgood, ed.), Academic Press, New York, 1973, pp. 129–199.
  19. [19] NIEDERREITER, H.: Random Number Generation and Quasi-Monte Carlo Methods. In: CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
  20. [20] OSTROMOUKHOV, V.: Recent progress in improvement of extreme discrepancy and star discrepancy of one-dimensional Sequences, In: Monte Carlo and Quasi-Monte Carlo Methods 2008, (P. L’Ecuyer, and A. B. Owen, eds.), Springer-Verlag, Berlin, 2009, pp. 561–572.10.1007/978-3-642-04107-5_36
  21. [21] PAUSINGER, F.: Weak multipliers for generalized van der Corput sequences, J. Théor. Nombres Bordeaux 24 (2012), no. 3, 729–749.
  22. [22] SCHMIDT, W. M.: Irregularities of distribution VII, Acta Arith. 21 (1972), 45–50.10.4064/aa-21-1-45-50
  23. [23] TOPUZOĞLU, A.: The Carlitz rank of permutations of finite fields: a survey, J. Symb. Comput. 64 (2014), 53–66.10.1016/j.jsc.2013.07.004
  24. [24] ZAREMBA, S. K.: La méthode des bons treillis pour le calcul des intégrals multiples. In: Applications of Number Theory to Numerical Analysis, (S. K. Zaremba, ed.), (Proc. Sympos., Univ. Montreal, Montreal, Que., 1971), Academic Press, New York, 1972, pp. 39–119.
DOI: https://doi.org/10.1515/udt-2018-0003 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X
Language: English
Page range: 47 - 64
Submitted on: Mar 31, 2017
Accepted on: Jul 10, 2017
Published on: Jul 20, 2018
Published by: Slovak Academy of Sciences, Mathematical Institute
In partnership with: Paradigm Publishing Services
Publication frequency: 2 times per year

© 2018 Florian Pausinger, Alev Topuzoğlu, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.