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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A construction related to the cosine problem


Author: Mihail N. Kolountzakis
Journal: Proc. Amer. Math. Soc. 122 (1994), 1115-1119
MSC: Primary 42A05; Secondary 68Q25
MathSciNet review: 1243831
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Abstract: We give a constructive proof of the fact that for any sequence of positive integers $ {n_1},{n_2}, \ldots ,{n_N}$ there is a subsequence $ {m_1}, \ldots ,{m_r}$ for which

$\displaystyle - \mathop {\min }\limits_x \sum\limits_1^r {\cos {m_j}x \geq CN,} $

where C is a positive constant. Uchiyama previously proved the above inequality with the right-hand side replaced by $ C\sqrt N $. We give a polynomial time algorithm for the selection of the subsequence $ {m_j}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1243831-8
PII: S 0002-9939(1994)1243831-8
Article copyright: © Copyright 1994 American Mathematical Society