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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}$.

References [Enhancements On Off] (What's this?)

  • [1] J. Bourgain, Sur le minimum d'une somme de cosinus, Acta Arith. 45 (1986), 381-389. MR 847298 (87g:11096)
  • [2] S. Chowla, Some applications of a method of A. Selberg, J. Reine Angew. Math. 217 (1965), 128-132. MR 0172853 (30:3070)
  • [3] A. M. Odlyzko, personal communication.
  • [4] S. Uchiyama, On the mean modulus of trigonometric polynomials whose coefficients have random signs, Proc. Amer. Math. Soc. 16 (1965), 1185-1190. MR 0185362 (32:2830)

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