Metric Discrepancy Results for Sequences {nkx} and Diophantine Equations

  • István Berkes
  • Walter Philipp
  • Robert F. Tichy
Part of the Developments in Mathematics book series (DEVM, volume 16)

Abstract

Let (n k ) be an increasing sequence of positive integers. For 0 ≤ x ≤ 1, set
$$ \eta _k = \eta _k \left( x \right): = n_k x \left( {mod 1} \right). $$
(1)

Keywords

Discrepancy lacunary sequences Diophantine equations law of iterated algorithm 

2000 Mathematics subject classification

11K38 11D45 42A55 60F15 

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References

  1. 1.
    Baker, R.C.: Metric number theory and the large sieve. J. Lond. Math. Soc. Ser. II 24, 34–40 (1981)MATHCrossRefGoogle Scholar
  2. 2.
    Berkes, I.: On almost i.i.d. subsequences of the trigonometric system. In: Odell, E.W., Rosenthal, H.P. (eds.) Functional Analysis: Proceedings of the Seminar at the University of Texas at Austin. Lect. Notes Math., vol. 1332, pp. 54–63. Springer, Heidelberg (1987)Google Scholar
  3. 3.
    Berkes, L, Philipp, W.: An a.s. invariance principle for lacunary series f (n k x). Acta Math. Acad. Sci. Hung. 34, 141–155 (1979)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Berkes, I., Philipp, W.: The size of trigonometric and Walsh series and uniform distribution mod 1. J. Lond. Math. Soc. Sen II. 50, 454–464 (1994)MATHMathSciNetGoogle Scholar
  5. 5.
    Cassels, J.W.S.: Some metrical theorems in Diophantine approximation III. Proc. Camb. Philos. Soc. 46, 219–225 (1950)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Drmota, M., Tichy, R.F.: Sequences, Discrepancies and Applications. Lect. Notes Math., vol. 1651. Springer, Heidelberg (1997)MATHGoogle Scholar
  7. 7.
    Erdős, P., Koksma, J.F.: On the uniform distribution modulo 1 of sequences (f (n,ϑ)). Proc. K. Ned. Akad. Wet. 52, 851–854 (1949)Google Scholar
  8. 8.
    Evertse, J.-H., Schlickewei, R.H.-R., Schmidt, W.M.: Linear equations in variables which lie in a multiplicative group. Ann. Math. (2) 155, 807–836 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fukuyama, K., Petit, B.: Le théorème limite central pour les suites de R. C. Baker. Ergodic Theory Dyn. Syst. 21, 479–492 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gaposhkin, V.F.: Lacunary series and independent functions (in Russian). Usp. Mat. Nauk 21(6), 3–82 (1966)MathSciNetGoogle Scholar
  11. 11.
    Gaposhkin, V.F.: On the central limit theorem for some weakly dependent sequences (in Russian). Teor. Verojatn. Primen. 15, 666–684 (1970)MATHGoogle Scholar
  12. 12.
    Kac, M.: On the distribution of values of sums of type Σ f (2 kt). Ann. Math. 47, 33–49 (1947)Google Scholar
  13. 13.
    Kac, M.: Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc. 55, 641–665 (1949)MATHCrossRefGoogle Scholar
  14. 14.
    Kahane, J.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985)MATHGoogle Scholar
  15. 15.
    Kesten, H.: The discrepancy of random sequences kx. ActaArith. 10, 183–213 (1964)MATHMathSciNetGoogle Scholar
  16. 16.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)MATHGoogle Scholar
  17. 17.
    Philipp, W.: Limit theorems for lacunary series and uniform distribution mod 1. ActaArith. 26, 241–251 (1975)MATHMathSciNetGoogle Scholar
  18. 18.
    Philipp, W.: A functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables. Ann. Probab. 5, 319–350 (1977)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Philipp, W.: Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory. Trans. Am. Math. Soc. 345, 707–727 (1994)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Schoissengeier, J.: A metrical result on the discrepancy of (). Glasg. Math. J. 40, 393–425 (1998)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Shorack, R., Wellner, J.: Empirical Processes with Applications to Statistics. Wiley, New York (1986)MATHGoogle Scholar
  22. 22.
    Tijdeman, R.: On integers with many small prime factors. Compos. Math. 26, 319–330 (1973)MATHMathSciNetGoogle Scholar
  23. 23.
    Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • István Berkes
    • 1
  • Walter Philipp
    • 2
  • Robert F. Tichy
    • 3
  1. 1.Institut für Mathematik ATechnische Universität GrazGrazAustria
  2. 2.Institut für StatistikTechnische Universität GrazGrazAustria
  3. 3.Department of StatisticsUniversity of Illinois at Urbana-ChampaignChampaignUSA

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