An optimal condition for the LIL for trigonometric series

Abstract:

By a classical theorem (Salem-Zygmund [6], Erdős-Gàl [3]), if $({n_k})$ is a sequence of positive integers satisfying ${n_{k + 1}}/{n_k} \geqslant q > 1\;(k = 1,2, \ldots )$ then $(\cos {n_k}x)$ obeys the law of the iterated logarithm, i.e., (1) \[ \lim \sup \limits _{N \to \infty } {(N\log \log N)^{ - 1/2}}\sum \limits _{k \leqslant N} {\cos {n_k}x = 1\quad {\text {a}}{\text {.e}}{\text {.}}} \] It is also known (Takahashi [7, 8]) that the Hadamard gap condition ${n_{k + 1}}/{n_k} \geqslant q > 1$ can be essentially weakened here but the problem of finding the precise gap condition for the LIL (1) has remained open. In this paper we find, using combinatorial methods, an optimal gap condition for the upper half of the LIL, i.e., the inequality $\leqslant 1$ in (1).