### The Law of the Iterated Logarithm for Lacunary Trigonometric Series

Transactions of the American Mathematical Society, Vol. 91, No. 3 (June 1959), 444-469

Introduction

The purpose of this paper is to prove the following theorem of "The Law of the Iterated Logarithm" for lacunary trigonometric series.

Theorem 1. *Let*

(1.1) \(\displaystyle S(x) = \sum_{k=0}^\infty \left(a_k \cos n_kx + b_k\sin n_kx\right)\)

*be a lacunary trigonometric series, that is to say, one such that n*_{k+1}/n_{k} > q > 1 for all k. We write\[B_N = \left( \frac{1}{2} \sum_{k=1}^N \left(a_k^2+b_k^2\right)\right)^{1/2}\]\[M_N = \max_{1 \le k \le N}\left( a_k^2+b_k^2 \right)^{1/2}\]\[S_N(x) = \sum_{k=0}^N \left( a_k \cos n_kx + b_k \sin n_kx\right)\]

*If, for \(N \rightarrow +\infty\)*,

(1.2) \(B_N \rightarrow \infty\) and \(\displaystyle M_N = o\left\{\frac{B_N}{\left(\log \log B_N\right)^{1/2}}\right\}\)

*then we have, for almost all x,*

(1.3) \(\displaystyle \limsup_{N \rightarrow +\infty}\frac{S_N(x)}{\left( 2B_N^2 \log \log B_N\right)^{1/2}} = 1.\)

In this theorem the *a*_{k} and *b*_{k} are real and the *n*_{k} are positive but not necessarily integral. The latter point is important for applications.

The Law of the Iterated Logarithm for lacunary series has already been treated in the literature. In the first place, Salem and Zygmund have shown that under the hypotheses (1.2) we have (1.3) with "__<__" instead of "=." They also presuppose that the *n*_{k} are integers. A complete proof of (1.3) was given later by Erdös and Gál under the restriction however that *S* is of the form \(\sum e^{in_kx}\) and that the *n*_{k} are integers. In the proof which follows we use some ideas from these two papers and also from the classical paper of Kolmogoroff on the Law of the Iterated Logarithm for independent random variables. More detailed acknowledgments will be given at the proper places.

I am grateful to Professor Zygmund for calling my attention to the problem and for helping me with suggestions.