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 ∝_{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