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

Open access

Abstract

A permuted van der Corput sequence Sbσ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. t(Sbσ):=limsupNDN(Sbσ)/logN is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for t(Spσ) for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that t(Spσ)<t(Spid).

[1] BOURGAIN, J.—KONTOROVICH, A.: On Zaremba’s conjecture, Ann. of Math. 180 (2014), no. 2. 1–6

[2] CARLITZ, L.: Permutations in a finite field, Proc. Amer. Math. Soc. 4 (1953), 538.

[3] CHAIX, H.—FAURE, H.: Discrépance et diaphonie en dimension un, Acta Arith. 63 (1993), 103–141.

[4] DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences. Cambridge Univ. Press, Cambridge, England, 2010.

[5] DRMOTA, M.—TICHY, R. F.: Sequences, Discrepancies and Applications. In: Lecture Notes in Math. Vol. 1651. Springer-Verlag, Berlin, 1997.

[6] FAURE, H.: Discrépance de suites associéesàunsystème de numération (en dimension un), Bull. Soc. Math. France 109 (1981), no 2, 143–182.

[7] FAURE, H.: Good permutations for extreme discrepancy, J. Number Theory 42 (1992), 47–56.

[8] FAURE, H.: Irregularities of distribution of digital (0, 1)-sequences in prime base, Integers 5 (2005), no. 3, A7, 12 pages.

[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] FAURE, H.: Star extreme discrepancy of generalized two-dimensional Hammersley point sets, Unif. Distrib. Theory 3 (2008), no. 2, 45–65.

[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.

[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] HUANG, S.: An Improvement to Zaremba’s Conjecture. Geometric and Functional Analysis 25 (2015), 860–914.

[14] KHINCHIN, A. YA.: Continued Fractions. The University of Chicago Press, Chicago, Ill.-London, 1964.

[15] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[16] LARCHER, G.: On the discrepancy of sequences in the unit-interval, Indag. Math., New Series 27 (2016), 546–558.

[17] MATOUŠEK, J.: On the L2-discrepancy for anchored boxes, J. Complexity 14 (1998), 527–556.

[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] 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] 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.

[21] PAUSINGER, F.: Weak multipliers for generalized van der Corput sequences, J. Théor. Nombres Bordeaux 24 (2012), no. 3, 729–749.

[22] SCHMIDT, W. M.: Irregularities of distribution VII, Acta Arith. 21 (1972), 45–50.

[23] TOPUZOĞLU, A.: The Carlitz rank of permutations of finite fields: a survey, J. Symb. Comput. 64 (2014), 53–66.

[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.

Uniform distribution theory

The Journal of Slovak Academy of Sciences

Journal Information


Mathematical Citation Quotient (MCQ) 2017: 0.30

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 119 119 5
PDF Downloads 73 73 5