Advertisement

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baker, R.C.: Metric number theory and the large sieve. J. Lond. Math. Soc. Ser. II 24, 34–40 (1981)zbMATHCrossRefGoogle 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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHMathSciNetGoogle 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)zbMATHGoogle 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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHGoogle 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)zbMATHCrossRefGoogle Scholar
  14. 14.
    Kahane, J.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  15. 15.
    Kesten, H.: The discrepancy of random sequences kx. ActaArith. 10, 183–213 (1964)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)zbMATHGoogle Scholar
  17. 17.
    Philipp, W.: Limit theorems for lacunary series and uniform distribution mod 1. ActaArith. 26, 241–251 (1975)zbMATHMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle 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)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Shorack, R., Wellner, J.: Empirical Processes with Applications to Statistics. Wiley, New York (1986)zbMATHGoogle Scholar
  22. 22.
    Tijdeman, R.: On integers with many small prime factors. Compos. Math. 26, 319–330 (1973)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916)zbMATHCrossRefMathSciNetGoogle 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

Personalised recommendations