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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An optimal condition for the LIL for trigonometric series

Author: I. Berkes
Journal: Trans. Amer. Math. Soc. 347 (1995), 515-530
MSC: Primary 42A32; Secondary 42A55
MathSciNet review: 1282883
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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)

$\displaystyle \mathop {\lim \sup }\limits_{N \to \infty } {(N\log \log N)^{ - 1... ...\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).

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Article copyright: © Copyright 1995 American Mathematical Society