A construction related to the cosine problem
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- by Mihail N. Kolountzakis PDF
- Proc. Amer. Math. Soc. 122 (1994), 1115-1119 Request permission
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 \[ - \min \limits _x \sum \limits _1^r {\cos {m_j}x \geq CN,} \] 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
- J. Bourgain, Sur le minimum d’une somme de cosinus, Acta Arith. 45 (1986), no. 4, 381–389 (French). MR 847298, DOI 10.4064/aa-45-4-381-389
- S. Chowla, Some applications of a method of A. Selberg, J. Reine Angew. Math. 217 (1965), 128–132. MR 172853, DOI 10.1515/crll.1965.217.128 A. M. Odlyzko, personal communication.
- S. Uchiyama, On the mean modulus of trigonometric polynomials whose coefficients have random signs, Proc. Amer. Math. Soc. 16 (1965), 1185–1190. MR 185362, DOI 10.1090/S0002-9939-1965-0185362-4
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 1115-1119
- MSC: Primary 42A05; Secondary 68Q25
- DOI: https://doi.org/10.1090/S0002-9939-1994-1243831-8
- MathSciNet review: 1243831