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  • 1 Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
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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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(n_k )_{k \geqq 1}$$ \end{document}
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2$$ \end{document}
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.

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