Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Christoph Aistleitner
Journal: Trans. Amer. Math. Soc. 362 (2010), 5967-5982.
MSC (2000): Primary 11K38, 42A55, 60F15
Posted: June 16, 2010
MathSciNet review: 2661504
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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.

References:

1.: C. Aistleitner and I. Berkes. On the central limit theorem for . Probab. Theory Relat. Fields, 146 (2010), no. 1-2, 267-289. MR 2550364
2.: R. C. Baker. Metric number theory and the large sieve. J. London. Math. Soc. 24, 34-40, 1981. MR 623668 (83a:10086)
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. 4:159-180, 1978. MR 514757 (80c:60032)
5.: I. Berkes and W. Philipp. An a.s. invariance principle for lacunary series . Acta Math. Acad. Sci. Hungar. 34:141-155, 1979. MR 546729 (80i:60042)
6.: I. Berkes and W. Philipp. The size of trigonometric and Walsh series and uniform distribution mod 1. J. London Math. Soc. (2), 50:454-464, 1994. MR 1299450 (96e:11099)
7.: J. W. S. Cassels. Some metrical theorems in Diophantine approximation III. Proc. Cambridge Philos. Soc. 46:219-225, 1950. MR 0036789 (12:162d)
8.: P. Erdős and J. F. Koksma. On the uniform distribution modulo of sequences $(f(n,\theta))$ . Proc. Kon. Nederl. Akad. Wetensch. 52:851-854, 1949. MR 0032690 (11:331f)
9.: K. Fukuyama. The law of the iterated logarithm for discrepancies of $\{\theta^n x\}$ . Acta Math. Hungar. 118:155-170, 2008. MR 2378547 (2008m:60049)
10.: V. F. Gaposhkin. Lacunary series and independent functions (in Russian). Uspehi Mat. Nauk 21/6:3-82,1966. MR 0206556 (34:6374)
11.: V. F. Gaposhkin. On the central limit theorem for some weakly dependent sequences (in Russian). Teor. Verojatn. Prim. 15:666-684, 1970. MR 0282394 (43:8106)
12.: M. Kac. On the distribution of values of sums of the type $\sum f(2^k t)$ . Ann. of Math. (2) 47:33-49, 1946. MR 0015548 (7:436f)
13.: M. Kac. Probability methods in some problems of analysis and number theory. Bull. Amer. Math. Soc. 55:641-665, 1949. MR 0031504 (11:161b)
14.: H. Kesten. The discrepancy of random sequences $\{kx\}$ , Acta Arith. 10:183-213, 1964/65. MR 0168546 (29:5807)
15.: L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. Wiley, New York, 1974. MR 0419394 (54:7415)
16.: W. Philipp. Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26:241-251, 1975. MR 0379420 (52:325)
17.: R. Shorack and J. Wellner. Empirical processes with applications to statistics. Wiley, New York, 1986. MR 838963 (88e:60002)
18.: V. Strassen. Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Vol. II: Contributions to Probability Theory:315-343, 1967. MR 0214118 (35:4969)
19.: S. Takahashi. An asymptotic property of a gap sequence. Proc. Japan Acad. 38:101-104, 1962. MR 0140865 (25:4279)
20.: H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77:313-352, 1916. MR 1511862
21.: A. Zygmund. Trigonometric series. Vols. I, II. Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2002. MR 1963498 (2004h:01041)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|