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

Aistleitner, Christoph

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

On the law of the iterated logarithm for the discrepancy of lacunary sequences
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Trans. Amer. Math. Soc. 362 (2010), 5967-5982 Request permission

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 $D_N$ of the sequence $(n_k x)_{k\ge 1}$ mod 1 satisfies the law of the iterated logarithm (LIL), i.e. \[ 1/4 \leq \limsup _{N \to \infty } N D_N(n_k x) (N \log \log N)^{-1/2} \leq C_q \quad \mathrm {a.e.}\] 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 $(n_k)$ 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.

References

Christoph Aistleitner and István Berkes, On the central limit theorem for $f(n_kx)$, Probab. Theory Related Fields 146 (2010), no. 1-2, 267–289. MR 2550364, DOI 10.1007/s00440-008-0190-6
R. C. Baker, Metric number theory and the large sieve, J. London Math. Soc. (2) 24 (1981), no. 1, 34–40. MR 623668, DOI 10.1112/jlms/s2-24.1.34
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.
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, DOI 10.1007/BF01908987
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, DOI 10.1007/BF01902603
István Berkes and Walter Philipp, The size of trigonometric and Walsh series and uniform distribution $\textrm {mod}\ 1$, J. London Math. Soc. (2) 50 (1994), no. 3, 454–464. MR 1299450, DOI 10.1112/jlms/50.3.454
J. W. S. Cassels, Some metrical theorems of Diophantine approximation. III, Proc. Cambridge Philos. Soc. 46 (1950), 219–225. MR 36789, DOI 10.1017/s0305004100025688
P. Erdös and J. F. Koksma, On the uniform distribution modulo $1$ of sequences $(f(n,\theta ))$, Nederl. Akad. Wetensch., Proc. 52 (1949), 851–854 = Indagationes Math. 11, 299–302 (1949). MR 32690
K. Fukuyama, The law of the iterated logarithm for discrepancies of $\{\theta ^nx\}$, Acta Math. Hungar. 118 (2008), no. 1-2, 155–170. MR 2378547, DOI 10.1007/s10474-007-6201-8
V. F. Gapoškin, Lacunary series and independent functions, Uspehi Mat. Nauk 21 (1966), no. 6 (132), 3–82 (Russian). MR 0206556
V. F. Gapoškin, The central limit theorem for certain weakly dependent sequences, Teor. Verojatnost. i Primenen. 15 (1970), 666–684 (Russian). MR 0282394
M. Kac, On the distribution of values of sums of the type $\sum f(2^k t)$, Ann. of Math. (2) 47 (1946), 33–49. MR 15548, DOI 10.2307/1969033
M. Kac, Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc. 55 (1949), 641–665. MR 31504, DOI 10.1090/S0002-9904-1949-09242-X
H. Kesten, The discrepancy of random sequences $\{kx\}$, Acta Arith. 10 (1964/65), 183–213. MR 168546, DOI 10.4064/aa-10-2-183-213
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
Walter Philipp, Limit theorems for lacunary series and uniform distribution $\textrm {mod}\ 1$, Acta Arith. 26 (1974/75), no. 3, 241–251. MR 379420, DOI 10.4064/aa-26-3-241-251
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
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. 315–343. MR 0214118
Shigeru Takahashi, An asymptotic property of a gap sequence, Proc. Japan Acad. 38 (1962), 101–104. MR 140865
Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864
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