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Irregular discrepancy behavior of lacunary series

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Abstract

In 1975 Philipp showed that for any increasing sequence (n k ) of positive integers satisfying the Hadamard gap condition n k+1/n k  > q > 1, k ≥ 1, the discrepancy D N of (n k x) mod 1 satisfies the law of the iterated logarithm

$$ 1/4 \leq {\mathop {\rm lim\,sup} \limits _{N\to\infty}}\, N D_N(n_k x) (N \log \log N)^{-1/2}\leq C_q\quad \textup{a.e.}$$

Recently, Fukuyama computed the value of the lim sup for sequences of the form n k = θk, θ > 1, and in a preceding paper the author gave a Diophantine condition on (n k ) for the value of the limsup to be equal to 1/2, the value obtained in the case of i.i.d. sequences. In this paper we utilize this number-theoretic connection to construct a lacunary sequence (n k ) for which the lim sup in the LIL for the star-discrepancy \({D_N^*}\) is not a constant a.e. and is not equal to the lim sup in the LIL for the discrepancy D N .

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References

  1. Aistleitner, C.: On the law of the iterated logarithm for the discrepancy of lacunary sequences. Trans. Am. Math. Soc. (2008, to appear)

  2. Aistleitner, C., Berkes, I.: On the central limit theorem for f(n k x). Prob. Theory Related Fields (2008, to appear)

  3. Berkes I.: On the central limit theorem for lacunary trigonometric series. Anal. Math. 4, 159–180 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berkes I., Philipp W.: An a.s. invariance principle for lacunary series f(n k x). Acta Math. Acad. Sci. Hungar. 34, 141–155 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fukuyama, K.: A law of the iterated logarithm for discrepancies: non-constant limsup. Monatsh. Math. (2008, to appear)

  6. Fukuyama K.: The law of the iterated logarithm for discrepancies of {θn x }. Acta Math. Hung. 118, 155–170 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kac M.: Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc. 55, z641–665 (1949)

    Article  Google Scholar 

  8. Kesten H.: The discrepancy of random sequences {kx}. Acta Arith. 10, 183–213 (1964/1965)

    MathSciNet  Google Scholar 

  9. Khinchin A.: Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math. Ann. 92, 115–125 (1924)

    Article  MathSciNet  Google Scholar 

  10. Philipp W.: Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26, 241–251 (1975)

    MATH  MathSciNet  Google Scholar 

  11. Shorack R., Wellner J.: Empirical Processes with Applications to Statistics. Wiley, New York (1986)

    MATH  Google Scholar 

  12. Strassen, V.: Almost sure behavior of sums of independent random variables and martingales. In: Proceedings of Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), vol. II. Contributions to Probability Theory, pp. 315–343 (1967)

  13. Takahashi S.: An asymptotic property of a gap sequence. Proc. Jpn. Acad. 38, 101–104 (1962)

    Article  MATH  Google Scholar 

  14. Weyl H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zygmund, A.: Trigonometric Series, vol. I, II. Reprint of the 1979 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988)

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Correspondence to Christoph Aistleitner.

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Communicated by J. Schoißengeier.

This research was supported by the Austrian Research Foundation (FWF), Project S9603-N13.

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Aistleitner, C. Irregular discrepancy behavior of lacunary series. Monatsh Math 160, 1–29 (2010). https://doi.org/10.1007/s00605-008-0067-x

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  • DOI: https://doi.org/10.1007/s00605-008-0067-x

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