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
for almost all x ∈ (0, 1), where ‖f‖2 = (∝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 ‖f‖2. 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 ‖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 (nk)k≧1 such that (1) holds with √‖f‖22 + g(x) instead of ‖f‖2 on the right-hand side.