Lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle
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- by S. V. Konyagin and W. Schlag
- Trans. Amer. Math. Soc. 351 (1999), 4963-4980
- DOI: https://doi.org/10.1090/S0002-9947-99-02241-2
- Published electronically: August 27, 1999
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Abstract:
Let $T(x)=\sum _{j=0}^{n-1}\pm e^{ijx}$ where $\pm$ stands for a random choice of sign with equal probability. The first author recently showed that for any $\epsilon >0$ and most choices of sign, $\min _{x\in [0,2\pi )}|T(x)|<n^{-1/2+\epsilon }$, provided $n$ is large. In this paper we show that the power $n^{-1/2}$ is optimal. More precisely, for sufficiently small $\epsilon >0$ and large $n$ most choices of sign satisfy $\min _{x\in [0,2\pi )}|T(x)|> \epsilon n^{-1/2}$. Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials.References
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Bibliographic Information
- S. V. Konyagin
- Affiliation: Institute for Advanced Study, School of Mathematics, Olden Lane, Princeton, New Jersey 08540
- Address at time of publication: Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia
- MR Author ID: 188475
- Email: kon@nw.math.msu.su
- W. Schlag
- Affiliation: Institute for Advanced Study, School of Mathematics, Olden Lane, Princeton, New Jersey 08540
- Address at time of publication: Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544
- MR Author ID: 313635
- Email: schlag@math.princeton.edu
- Received by editor(s): February 5, 1997
- Received by editor(s) in revised form: September 24, 1997
- Published electronically: August 27, 1999
- Additional Notes: The authors were supported by the National Science Foundation, grant DMS 9304580. This research was carried out while the authors were members of the Institute for Advanced Study, Princeton. It is a pleasure to thank the Institute for its hospitality and generosity. The authors would like to thank A. G. Karapetian for comments on a preliminary version of this paper.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 4963-4980
- MSC (1991): Primary 42A05, 42A61; Secondary 30C15, 60F05
- DOI: https://doi.org/10.1090/S0002-9947-99-02241-2
- MathSciNet review: 1487621