On the law of the iterated logarithm for the discrepancy of lacunary sequences

Aistleitner, Christoph

doi:10.1090/S0002-9947-2010-05026-3

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ISSN 1088-6850(online) ISSN 0002-9947(print)

On the law of the iterated logarithm for the discrepancy of lacunary sequences

Author: Christoph Aistleitner
Journal: Trans. Amer. Math. Soc. 362 (2010), 5967-5982
MSC (2000): Primary 11K38, 42A55, 60F15
Posted: June 16, 2010
MathSciNet review: 2661504
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Abstract: A classical result of Philipp (1975) states that for any sequence $(n_k)_{k \geq 1}$ of integers satisfying the Hadamard gap condition $n_{k+1}/n_k\ge q>1 (k=1, 2, \ldots)$ , the discrepancy of the sequence $(n_k x)_{k\ge 1}$ mod 1 satisfies the law of the iterated logarithm (LIL), i.e.

$\displaystyle 1/4 \leq \limsup_{N \to \infty} N D_N(n_k x) (N \log \log N)^{-1/2} \leq C_q \quad {a.e.}$

The value of the $\limsup$ is a long-standing open problem. Recently Fukuyama explicitly calculated the value of the $\limsup$ for $n_k= \theta^k$ , $\theta>1$ , not necessarily integer. We extend Fukuyama's result to a large class of integer sequences

characterized in terms of the number of solutions of a certain class of Diophantine equations and show that the value of the $\limsup$ is the same as in the Chung-Smirnov LIL for i.i.d. random variables.

1. Christoph Aistleitner and István Berkes, On the central limit theorem for 𝑓(𝑛_{𝑘}𝑥), Probab. Theory Related Fields 146 (2010), no. 1-2, 267–289. MR 2550364 (2010i:42015), http://dx.doi.org/10.1007/s00440-008-0190-6
2. R. C. Baker, Metric number theory and the large sieve, J. London Math. Soc. (2) 24 (1981), no. 1, 34–40. MR 623668 (83a:10086), http://dx.doi.org/10.1112/jlms/s2-24.1.34
3. I. Berkes. On the asymptotic behaviour of $\sum f(n_k x)$ . I. Main theorems, II. Applications. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 34:319-345; 347-365, 1976.
4. I. Berkes, On the central limit theorem for lacunary trigonometric series, Anal. Math. 4 (1978), no. 3, 159–180 (English, with Russian summary). MR 514757 (80c:60032), http://dx.doi.org/10.1007/BF01908987
5. I. Berkes and W. Philipp, An a.s. invariance principle for lacunary series 𝑓(𝑛_{𝑘}𝑥), Acta Math. Acad. Sci. Hungar. 34 (1979), no. 1-2, 141–155. MR 546729 (80i:60042), http://dx.doi.org/10.1007/BF01902603
6. István Berkes and Walter Philipp, The size of trigonometric and Walsh series and uniform distribution 𝑚𝑜𝑑1, J. London Math. Soc. (2) 50 (1994), no. 3, 454–464. MR 1299450 (96e:11099), http://dx.doi.org/10.1112/jlms/50.3.454
7. J. W. S. Cassels, Some metrical theorems of Diophantine approximation. III, Proc. Cambridge Philos. Soc. 46 (1950), 219–225. MR 0036789 (12,162d)
8. P. Erdös and J. F. Koksma, On the uniform distribution modulo 1 of sequences (𝑓(𝑛,𝜃)), Nederl. Akad. Wetensch., Proc. 52 (1949), 851–854 = Indagationes Math. 11, 299–302 (1949). MR 0032690 (11,331f)
9. K. Fukuyama, The law of the iterated logarithm for discrepancies of {𝜃ⁿ𝑥}, Acta Math. Hungar. 118 (2008), no. 1-2, 155–170. MR 2378547 (2008m:60049), http://dx.doi.org/10.1007/s10474-007-6201-8
10. V. F. Gapoškin, Lacunary series and independent functions, Uspehi Mat. Nauk 21 (1966), no. 6 (132), 3–82 (Russian). MR 0206556 (34 #6374)
11. V. F. Gapoškin, The central limit theorem for certain weakly dependent sequences, Teor. Verojatnost. i Primenen. 15 (1970), 666–684 (Russian). MR 0282394 (43 #8106)
12. M. Kac, On the distribution of values of sums of the type ∑𝑓(2^{𝑘}𝑡), Ann. of Math. (2) 47 (1946), 33–49. MR 0015548 (7,436f)
13. M. Kac, Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc. 55 (1949), 641–665. MR 0031504 (11,161b), http://dx.doi.org/10.1090/S0002-9904-1949-09242-X
14. H. Kesten, The discrepancy of random sequences {𝑘𝑥}, Acta Arith. 10 (1964/1965), 183–213. MR 0168546 (29 #5807)
15. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York, 1974. Pure and Applied Mathematics. MR 0419394 (54 #7415)
16. Walter Philipp, Limit theorems for lacunary series and uniform distribution 𝑚𝑜𝑑1, Acta Arith. 26 (1974/75), no. 3, 241–251. MR 0379420 (52 #325)
17. Galen R. Shorack and Jon A. Wellner, Empirical processes with applications to statistics, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986. MR 838963 (88e:60002)
18. Volker Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Univ. California Press, Berkeley, Calif., 1967, pp. Vol. II: Contributions to Probability Theory, Part 1, pp. 315–343. MR 0214118 (35 #4969)
19. Shigeru Takahashi, An asymptotic property of a gap sequence, Proc. Japan Acad. 38 (1962), 101–104. MR 0140865 (25 #4279)
20. Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, http://dx.doi.org/10.1007/BF01475864
21. A. Zygmund, Trigonometric series. Vol. I, II, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. With a foreword by Robert A. Fefferman. MR 1963498 (2004h:01041)

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