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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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An optimal condition for the LIL for trigonometric series
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by I. Berkes PDF
Trans. Amer. Math. Soc. 347 (1995), 515-530 Request permission

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).
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 515-530
  • MSC: Primary 42A32; Secondary 42A55
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1282883-1
  • MathSciNet review: 1282883