Skip to main content
Log in

On the class of limits of lacunary trigonometric series

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Let (n k )k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n k+1/n k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝ 10 f(x) dx = 0. Then the probabilistic behavior of the system (f(n k x))k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erdős and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary \( (n_k )_{k \geqq 1} \):

$$ \mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2 $$
((1))

for almost all x ∈ (0, 1), where ‖f2 = (∝ 10 f(x)2 dx)1/2 is the standard deviation of the random variables f(n k x). If (n k )k≧1 has certain number-theoretic properties (e.g. n k+1/n k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f2. For general lacunary (n k )k≧1 this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n k )k≧1, such that the lim sup in the LIL (1) is not equal to ‖f2 and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f such that for any function g(x) with sufficiently small Fourier coefficients there exists a lacunary sequence (n k )k≧1 such that (1) holds with √‖f 22 + g(x) instead of ‖f2 on the right-hand side.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Aistleitner, Diophantine equations and the LIL for the discrepancy of sub-lacunary sequences, Illinois J. Math., to appear.

  2. C. Aistleitner, Irregular discrepancy behavior of lacunary series, Monatshefte Math., to appear.

  3. C. Aistleitner, Irregular discrepancy behavior of lacunary series II, Monatshefte Math., to appear.

  4. C. Aistleitner, On the law of the iterated logarithm for the discrepancy of lacunary sequences, Trans. Amer. Math. Soc., to appear.

  5. C. Aistleitner and I. Berkes, On the central limit theorem for f(nkx), Probab. Theory Related Fields, 146 (2010), 267–289.

    Article  MathSciNet  Google Scholar 

  6. J. Arias de Reyna, Pointwise Convergence of Fourier Series, Springer-Verlag (Berlin, 2002).

    MATH  Google Scholar 

  7. I. Berkes, A central limit theorem for trigonometric series with small gaps, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47 (1979), 157–161.

    Article  MATH  MathSciNet  Google Scholar 

  8. I. Berkes and W. Philipp, An a.s. invariance principle for lacunary series f(nkx), Acta Math. Acad. Sci. Hung., 34 (1979), 141–155.

    Article  MATH  MathSciNet  Google Scholar 

  9. I. Berkes, W. Philipp and R. F. Tichy, Empirical processes in probabilistic number theory: The LIL for the discrepancy of (nkω) mod 1, Illinois J. Math., 50 (2008), 107–145.

    MathSciNet  Google Scholar 

  10. I. Berkes, W. Philipp and R. F. Tichy, Metric discrepancy results for sequences {n k x} and diophantine equations, Diophantine Equations. Festschrift für Wolfgang Schmidt. Developments in Mathematics 17, Springer, pp. 95–105.

  11. P. Erdős and I. S. Gál, On the law of the iterated logarithm, Proc. Kon. Nederl. Akad. Wetensch., 58 (1955), 65–84.

    Google Scholar 

  12. K. Fukuyama, A law of the iterated logarithm for discrepancies: non-constant limsup, Monatshefte Math., to appear.

  13. K. Fukuyama, The law of the iterated logarithm for discrepancies of {θ n x}, Acta Math. Hungar., 118 (2008), 155–170.

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Fukuyama and K. Nakata, A metric discrepancy result for the Hardy-Littlewood-Pólya sequences, Monatshefte Math., to appear.

  15. K. Fukuyama and S. Takahashi, On limit distributions of trigonometric sums, Rev. Roumaine Math. Pures Appl., 53 (2008), 19–24.

    MATH  MathSciNet  Google Scholar 

  16. V. F. Gaposhkin, Lacunary series and independent functions, Russian Math. Surveys, 21 (1966), 3–82.

    Article  Google Scholar 

  17. M. Kac, Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc., 55 (1949), 641–665.

    Article  MATH  MathSciNet  Google Scholar 

  18. C. J. Mozzochi, On the Pointwise Convergence of Fourier Series, Springer-Verlag (Berlin-New York, 1971).

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  20. R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad. Sci. USA, 33 (1947), 333–338.

    Article  MATH  MathSciNet  Google Scholar 

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

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

    Article  MATH  MathSciNet  Google Scholar 

  23. S. Takahashi, The law of the iterated logarithm for a gap sequence with infinite gaps, Tohoku Math. J., 15 (1963), 281–288.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to C. Aistleitner.

Additional information

Research supported by the Austrian Research Foundation (FWF), Project S9603-N23. This paper was written during a stay at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences, which was made possible by an MOEL scholarship of the Österreichische Forschungsgemeinschaft (ÖFG).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aistleitner, C. On the class of limits of lacunary trigonometric series. Acta Math Hung 129, 1–23 (2010). https://doi.org/10.1007/s10474-010-9218-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-010-9218-3

Key words and phrases

2000 Mathematics Subject Classification

Navigation