On the law of the iterated logarithm for the discrepancy of lacunary sequences II

Abstract:

By a classical heuristics, lacunary function systems exhibit many asymptotic properties which are typical for systems of independent random variables. For example, for a large class of functions $f$ the system $(f(n_k x))_{k \geq 1}$, where $(n_k)_{k \geq 1}$ is a lacunary sequence of integers, satisfies a law of the iterated logarithm (LIL) of the form \begin{equation}c_1 \leq \limsup _{N \to \infty } \frac {\sum _{k=1}^N f(n_k x)}{\sqrt {2 N \log \log N}} \leq c_2 \qquad \mathrm {a.e.}, \end{equation} where $c_1,c_2$ are appropriate positive constants. In a previous paper we gave a criterion, formulated in terms of the number of solutions of certain linear Diophantine equations, which guarantees that the value of the $\limsup$ in (1) equals the $L^2$-norm of $f$ for $\mathrm {a.e.}$ $x$, which is exactly what one would also expect in the case of i.i.d. random variables. This result can be used to prove a precise LIL for the discrepancy of $(n_k x)_{k \geq 1}$, which corresponds to the Chung-Smirnov LIL for the Kolmogorov-Smirnov-statistic of i.i.d. random variables.

In the present paper we give a full solution of the problem in the case of “stationary” Diophantine behavior, by this means providing a unifying explanation of the aforementioned “regular” LIL behavior and the “irregular” LIL behavior which has been observed by Kac, Erdős, Fortet and others.