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A lower bound on a quantity related to the quality of polynomial lattices. (English) Zbl 1247.11097
For \(b\) be a prime, let \(\mathbb{F}_b\) be a finite field consisting of \(b\) elements. Let \(f\in \mathbb{F}_b[x]\) be a polynomial over \(\mathbb{F}_b\). For a vector \(\mathbf{g}=(g_1,\ldots, g_s)\in\mathbb{F}_b[x]^s\), let \(P(\mathbf{g},f)\) be a polynomial lattice with \(N=b^m\) points in dimension \(s\). Let \(G_{b,m}=\{a\in\mathbb{F}_b[x]: \deg(a)<m\}\). A quantity \(R_b(\mathbf{g},f)\) is used for studying the quality of \(P(\mathbf{g},f)\). It was shown by H. Niederreiter [Random number generation and quasi-Monte Carlo methods. Philadelphia, PA: SIAM (1992; Zbl 0761.65002)] that the star discrepancy of \(P(\mathbf{g},f)\) satisfies \(D_N^*(P(\mathbf{g},f))\leq s/N+R_b(\mathbf{g},f)\) and there exists a vector \(\mathbf{g}\in G_{b,m}^s\) such that \(R_b(\mathbf{g},f)\leq C_{s,b}m^s/b^m\) for some \(C_{s,b}>0\). These results yield the existence of \(P(\mathbf{g},f)\) with \(D_N^*(P(\mathbf{g},f))=O((\log N)^s/N)\). In the paper under review, the authors prove that Niederreiter’s upper bound for \(R_b(\mathbf{g},f)\) is essentially best possible.

MSC:
11K06 General theory of distribution modulo \(1\)
11K38 Irregularities of distribution, discrepancy
Citations:
Zbl 0761.65002
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References:
[1] J. Dick, P. Kritzer, G. Leobacher, F. Pillichshammer, Constructions of general polynomial lattice rules based on the weighted star discrepancy , Finite Fields Appl. 13 (2007), 1045-1070. · Zbl 1132.11039
[2] J. Dick, F. Y. Kuo, F. Pillichshammer, I. H. Sloan, Construction algorithms for polynomial lattice rules for multivariate integration , Math. Comp. 74 (2005), 1895-1921. · Zbl 1079.65007
[3] J. Dick, G. Leobacher, F. Pillichshammer, Construction algorithms for digital nets with small weighted star discrepancy , SIAM J. Numer. Anal. 43 (2005), 76-95. · Zbl 1084.11040
[4] J. Dick, F. Pillichshammer, Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces , J. Complexity 21 (2005), 149-195. · Zbl 1085.41021
[5] J. Dick, F. Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration , Cambridge University Press, 2010. · Zbl 1282.65012
[6] M. Drmota, R. F. Tichy, Sequences, Discrepancies and Applications , Springer, Berlin, 1997. · Zbl 0877.11043
[7] M. Fuchs, Kettenbrüche im Körper der formalen Laurentreihen und Anwendungen , Master thesis, Vienna University of Technology, 2000 (available at http://dmg.tuwien.ac.at/drmota/ (state: August 17, 2011)). · Zbl 1016.68105
[8] L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences . John Wiley, New York, 1974. · Zbl 0281.10001
[9] G. Larcher, On the distribution of sequences connected with good lattice points , Monatsh. Math. 101 (1986), 135-150. · Zbl 0584.10030
[10] G. Larcher, A best lower bound for good lattice points , Monatsh. Math. 104 (1987), 45-51. · Zbl 0624.10043
[11] G. Larcher, Digital point sets: analysis and applications , in: Random and Quasi-Random Point Sets, P. Hellekalek and G. Larcher (eds.), Lecture Notes in Statistics Vol. 138, Springer, New York, 1998, 167-222. · Zbl 0920.11055
[12] G. Larcher, A. Lauss, H. Niederreiter, W. Ch. Schmid, Optimal polynomials for \((t,m,s)\)-nets and numerical integration of multivariate Walsh series , SIAM J. Numer. Anal. 33 (1996), 2239-2253. · Zbl 0861.65019
[13] P. L’Ecuyer, Polynomial integration lattices , in: Monte Carlo and Quasi-Monte Carlo Methods 2002, H. Niederreiter (ed.), Springer, Berlin, 2004, 73-98. · Zbl 1041.65008
[14] C. Lemieux, P. L’Ecuyer, Randomized polynomial lattice rules for multivariate integration and simulation , SIAM J. Sci. Comput. 24 (2003), 1768-1789. · Zbl 1071.11049
[15] H. Niederreiter, Point sets and sequences with small star discrepancy . Monatsh. Math. 104 (1987), 273-337. · Zbl 0626.10045
[16] H. Niederreiter, Sequences with almost perfect linear complexity profile , in: Advances in Cryptology–EUROCRYPT ’87, D. Chaum and W.L. Price (eds.), Lecture Notes in Computer Science Vol. 304, Springer, Berlin, 1988, 37-51. · Zbl 0651.94003
[17] H. Niederreiter, Low discrepancy point sets obtained by digital constructions over finite fields . Czechoslovak Math. J. 42 (1992), 143-166. · Zbl 0757.11024
[18] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods , SIAM, Philadelphia, 1992. · Zbl 0761.65002
[19] I. H. Sloan, S. Joe, Lattice methods for multiple integration , Oxford University Press, Oxford, 1994. · Zbl 0855.65013
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