A lower bound in the tail law of the iterated logarithm for lacunary trigonometric series

Abstract:

Salem and Zygmund obtained an upper bound for a tail law of the iterated logarithm for sums of the form $\sum _{k=N}^{\infty } a_{k} \cos (n_{k}x)+b_k \sin (n_{k}x)$, where $n_{k}$ satisfies a Hadamard gap condition and $\sum _{k=1}^{\infty } a_k^2 + b_k^2 < \infty .$ Here we obtain a lower bound in their result under the same hypotheses.