Acta Mathematica Hungarica

, Volume 129, Issue 1–2, pp 1–23

# On the class of limits of lacunary trigonometric series

Article

## Abstract

Let (nk)k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying nk+1/nk > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝01f(x) dx = 0. Then the probabilistic behavior of the system (f(nkx))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 = (∝01f(x)2dx)1/2 is the standard deviation of the random variables f(nkx). If (nk)k≧1 has certain number-theoretic properties (e.g. nk+1/nk → ∞), 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 (nk)k≧1 this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (nk)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 (nk)k≧1 such that (1) holds with √‖f22 + g(x) instead of ‖f2 on the right-hand side.

### Key words and phrases

lacunary series law of the iterated logarithm

### 2000 Mathematics Subject Classification

primary 11K38, 42A55, 60F15

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### References

1. [1]
C. Aistleitner, Diophantine equations and the LIL for the discrepancy of sub-lacunary sequences, Illinois J. Math., to appear.Google Scholar
2. [2]
C. Aistleitner, Irregular discrepancy behavior of lacunary series, Monatshefte Math., to appear.Google Scholar
3. [3]
C. Aistleitner, Irregular discrepancy behavior of lacunary series II, Monatshefte Math., to appear.Google Scholar
4. [4]
C. Aistleitner, On the law of the iterated logarithm for the discrepancy of lacunary sequences, Trans. Amer. Math. Soc., to appear.Google Scholar
5. [5]
C. Aistleitner and I. Berkes, On the central limit theorem for f(nkx), Probab. Theory Related Fields, 146 (2010), 267–289.
6. [6]
J. Arias de Reyna, Pointwise Convergence of Fourier Series, Springer-Verlag (Berlin, 2002).
7. [7]
I. Berkes, A central limit theorem for trigonometric series with small gaps, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47 (1979), 157–161.
8. [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.
9. [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.
10. [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.Google Scholar
11. [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. [12]
K. Fukuyama, A law of the iterated logarithm for discrepancies: non-constant limsup, Monatshefte Math., to appear.Google Scholar
13. [13]
K. Fukuyama, The law of the iterated logarithm for discrepancies of {θ n x}, Acta Math. Hungar., 118 (2008), 155–170.
14. [14]
K. Fukuyama and K. Nakata, A metric discrepancy result for the Hardy-Littlewood-Pólya sequences, Monatshefte Math., to appear.Google Scholar
15. [15]
K. Fukuyama and S. Takahashi, On limit distributions of trigonometric sums, Rev. Roumaine Math. Pures Appl., 53 (2008), 19–24.
16. [16]
V. F. Gaposhkin, Lacunary series and independent functions, Russian Math. Surveys, 21 (1966), 3–82.
17. [17]
M. Kac, Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc., 55 (1949), 641–665.
18. [18]
C. J. Mozzochi, On the Pointwise Convergence of Fourier Series, Springer-Verlag (Berlin-New York, 1971).
19. [19]
W. Philipp, Limit theorems for lacunary series and uniform distribution mod 1, Acta Arith., 26 (1975), 241–251.
20. [20]
R. Salem and A. Zygmund, On lacunary trigonometric series, Proc. Nat. Acad. Sci. USA, 33 (1947), 333–338.
21. [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.Google Scholar
22. [22]
S. Takahashi, An asymptotic property of a gap sequence, Proc. Japan Acad., 38 (1962), 101–104.
23. [23]
S. Takahashi, The law of the iterated logarithm for a gap sequence with infinite gaps, Tohoku Math. J., 15 (1963), 281–288.