# On the class of limits of lacunary trigonometric series

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## Abstract

Let ( for almost all

*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 ∝_{0}^{1}*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 2*πx*and lacunary \( (n_k )_{k \geqq 1} \):$$
\mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2
$$

(1)

*x*∈ (0, 1), where ‖*f*‖_{2}= (∝_{0}^{1}*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*‖_{2}^{2}+*g*(*x*) instead of ‖*f*‖_{2}on the right-hand side.### Key words and phrases

lacunary series law of the iterated logarithm### 2000 Mathematics Subject Classification

primary 11K38, 42A55, 60F15## Preview

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