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On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

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Abstract

In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ⊂ ℝs, we construct Kronecker-like and van der Corput-like ergodic transformations T 1,Γ and T 2,Γ of [0, 1)s. We prove that for admissible lattices Γ, (T nν,Γ (x))n≥0 is a low discrepancy sequence for all x ∈ [0, 1)s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ⊂ [0, 1)s, for almost all lattices Γ ∈ L s = SL(s,ℝ)/SL(s, ℤ) (in the sense of the invariant measure on L s ), the following asymptotic formula

$$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$$

holds with arbitrary small ɛ > 0, for all x ∈ [0, 1)s, and ν ∈ {1, 2}.

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References

  1. R. C. Baker, On irregularities of distribution. II, Journal of the London Mathematical Society. Second Series 59 (1999), 50–64.

    MATH  MathSciNet  Google Scholar 

  2. J. Beck, Probabilistic Diophantine approximation, I. Kronecker sequences, Annals of Mathematics. Second Series 140 (1994), 109–160.

    Google Scholar 

  3. J. Beck, Randomness of n√2 mod 1 and a Ramsey property of the hyperbola, in Sets, Graphs and Numbers (Budapest, 1991), Colloq. Math. Soc. Janos Bolyai, 60, North-Holland, Amsterdam, 1992, pp. 23–66.

    Google Scholar 

  4. J. Beck, Randomness in lattice point problems, Combinatorics, graph theory, algorithms and applications. Discrete Mathematics 229 (2001), 29–55.

    MATH  Google Scholar 

  5. J. Beck and W. W. L Chen, Irregularities of Distribution, Cambridge University Press, Cambridge, 1987.

    Book  MATH  Google Scholar 

  6. J. P. Borel, Suites dont la discrépance est comparable à un logarithme, Monatshefte für Mathematik 110 (1990), 207–216.

    Article  MATH  MathSciNet  Google Scholar 

  7. V. A. Bykovskiĭ, Estimates for the deviations of optimal grids in the L p-norm and the theory of quadrature formulas, Analysis Mathematica 22 (1996), 81–97.

    Article  MATH  MathSciNet  Google Scholar 

  8. V. A. Bykovskiĭ, On the right order of error of optimal cubature formulas in the spaces with dominanting derivation and L 2 discrepancy of nets, Dalnevost, Science Center of the USSR Acad. of Sciences, Vladivostok, 1985 (in Russian).

    Google Scholar 

  9. N. M. Dobrovolsky, Number-theoretic nets and their applications, Diss. kand. phi.-math. nauk, Tulsk. Gos. Ped. Inst., Tula, 1984 (in Russian).

    Google Scholar 

  10. M. Drmota and R. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Mathematics 1651, 1997.

  11. J. H. Evertse, H. P. Schlickewei and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, Annals of Mathematics. Second Series 155 (2002), 807–836.

    MATH  MathSciNet  Google Scholar 

  12. H. Faure, Discrépance de suites associées à un système de numération (en dimension s), Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica 41 (1982), 337–351.

    MATH  MathSciNet  Google Scholar 

  13. N. A. Friedman, Introduction to Ergodic Theory, Van Nostrand Reinhold Co., New-York, 1970.

    MATH  Google Scholar 

  14. K. K. Frolov, Upper bounds for the errors of quadrature formulae on classes of functions, Doklady Akademii Nauk SSSR 231 (1976), 818–821; English translation: Soviet Math. Dokl. 17 (1976), 1665–1669 (1977).

    MathSciNet  Google Scholar 

  15. K. K. Frolov, Quadrature formulas on classes of functions, Diss. kand. phi.-math. nauk, VC AN SSSR, Moskow, 1979 (in Russian).

    Google Scholar 

  16. K. K. Frolov, Upper bound of the discrepancy in metric L p, 2 ≤ p < ∞, (Russian) Doklady Akademii Nauk SSSR 252 (1980), no. 4, 805–807; English translation: Soviet Math. Dokl. 21 (1980), 840–842 (1981).

    MathSciNet  Google Scholar 

  17. P. M. Gruber and C. F. Lekkerkerker, Geometry of Numbers, North-Holland, Amsterdam, 1987.

    MATH  Google Scholar 

  18. A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms, Ergodic Theory and Dynamical Systems 15 (1995), 569–592.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. Krätzel, Lattice Points, VEB, Berlin, 1988.

    MATH  Google Scholar 

  20. L. Kuipers and H. Niedrreiter, Uniform Distribution of Sequences, John Wiley, New York, 1974.

    MATH  Google Scholar 

  21. V. F. Lev, The exact order of generalized diaphony and multidimensional numerical integration, Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics 66 (1999), 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. B. Levin, On the low discrepancy sequences and the generators of pseudorandom numbers, in Abstract of Conference on Algebraic Number Theory and Diophantine Analysis, Graz, Austria, 30.08–4.09, 1998, pp. 40–41.

  23. M. B. Levin, Randomness in lattice point problems and in low discrepancy sequences, preprint.

  24. M. B. Levin, Adelic construction of low discrepancy sequences, Online Journal of Analytic Combinatoric, to appear.

  25. M. B. Levin and E. Merzbach, Central limit theorems for the ergodic adding machine, Israel Journal of Mathematics 134 (2003), 61–92.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Matoušek, Geometric Discrepancy. An illustrated guide, Springer-Verlag, Berlin, 1999.

    MATH  Google Scholar 

  27. M. Mori, Construction of two dimensional low discrepancy sequences, Monte Carlo Methods and Applications 8 (2002), 159–169.

    Article  MATH  MathSciNet  Google Scholar 

  28. H. Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMSNSF Regional Conference Series in Applied Mathematics — 63, Philadelphia, 1992.

  29. S. Ninomiya, On the discrepancy of the β-adic van der Corput sequence, The University of Tokyo. Journal of Mathematical Sciences 5 (1998), 345–366.

    MATH  MathSciNet  Google Scholar 

  30. G. Pages, Van der Corput sequences, Kakutani transforms and one-dimensional numerical integration, Journal of Computational and Applied Mathematics 44 (1992),21–39.

    Article  MATH  MathSciNet  Google Scholar 

  31. K. Petersen, Ergodic Theory, Cambridge University Press, Cambridge, 1983.

    MATH  Google Scholar 

  32. M. M. Skriganov, Construction of uniform distributions in terms of geometry of numbers, Algebra i Analiz, 6, no. 3 (1994), 200–230; Reprinted in St. Petersburg Math. J., 6, no. 3 (1995), 635–664.

    MATH  MathSciNet  Google Scholar 

  33. M.M. Skriganov, Ergodic theory on SL(n), diophantine approximations and anomalies in the lattice point problem, Inventiones Mathematicae 132 (1998), 1–72.

    Article  MATH  MathSciNet  Google Scholar 

  34. M. M. Skriganov and A. N. Starkov On logarithmically small errors in the lattice point problem, Ergodic Theory and Dynamical Systems, 20 (2000), 1469–1476.

    Article  MATH  MathSciNet  Google Scholar 

  35. I. M. Sobol, On the distribution of points in a cube and the approximate evaluation of integrals, USSR Comput. Math. and Math. Phys., 1967, vol. 7,no. 4, pp. 86–112.

    Article  MATH  MathSciNet  Google Scholar 

  36. I. M. Sobol, Multidimensional quadrature formulas and Haar functions, Izdat. “Nauka”, Moskow, 1969. (in Russian).

    Google Scholar 

  37. R. Salem and A. Zygmund, On lacunary trigonometric series, Proceedings of the National Academy of Sciences of the United States of America 33, (1947). 333–338.

    Article  MATH  MathSciNet  Google Scholar 

  38. V. N. Temlyakov, Approximation of Periodic Function, Nova Science Publishers Inc., New York, 1993. 107–112

    Google Scholar 

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  1. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Israel

    Mordechay B. Levin

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Levin, M.B. On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube. Isr. J. Math. 178, 61–106 (2010). https://doi.org/10.1007/s11856-010-0058-1

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