Abstract
Let (n k )k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n k+1/n k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝ 10 f(x) dx = 0. Then the probabilistic behavior of the system (f(n k x))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 \( (n_k )_{k \geqq 1} \):
for almost all x ∈ (0, 1), where ‖f‖2 = (∝ 10 f(x)2 dx)1/2 is the standard deviation of the random variables f(n k x). If (n k )k≧1 has certain number-theoretic properties (e.g. n k+1/n k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f‖2. For general lacunary (n k )k≧1 this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n k )k≧1, such that the lim sup in the LIL (1) is not equal to ‖f‖2 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 (n k )k≧1 such that (1) holds with √‖f‖ 22 + g(x) instead of ‖f‖2 on the right-hand side.
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Research supported by the Austrian Research Foundation (FWF), Project S9603-N23. This paper was written during a stay at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences, which was made possible by an MOEL scholarship of the Österreichische Forschungsgemeinschaft (ÖFG).
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Aistleitner, C. On the class of limits of lacunary trigonometric series. Acta Math Hung 129, 1–23 (2010). https://doi.org/10.1007/s10474-010-9218-3
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DOI: https://doi.org/10.1007/s10474-010-9218-3