Lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle

Konyagin, S.; Schlag, W.

doi:10.1090/S0002-9947-99-02241-2

Lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle
HTML articles powered by AMS MathViewer

by S. V. Konyagin and W. Schlag PDF

Trans. Amer. Math. Soc. 351 (1999), 4963-4980 Request permission

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

R. N. Bhattacharya and R. Ranga Rao, Normal approximation and asymptotic expansions, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1976. MR 0436272
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
A. G. Karapetian. On the minimum of the absolute value of trigonometric polynomials with random coefficients. (Russian) Mat. Zametki 61 (1997), no. 3, 451–455.
B. S. Kashin, The properties of random trigonometric polynomials with $\pm 1$ coefficients, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1987), 40–46, 105 (Russian). MR 913269
S. V. Konyagin, On the minimum modulus of random trigonometric polynomials with coefficients $\pm 1$, Mat. Zametki 56 (1994), no. 3, 80–101, 158 (Russian, with Russian summary); English transl., Math. Notes 56 (1994), no. 3-4, 931–947 (1995). MR 1309842, DOI 10.1007/BF02362411
J. E. Littlewood, On polynomials $\sum ^{n}\pm z^{m}$, $\sum ^{n}e^{\alpha _{m}i}z^{m}$, $z=e^{\theta _{i}}$, J. London Math. Soc. 41 (1966), 367–376. MR 196043, DOI 10.1112/jlms/s1-41.1.367
Larry A. Shepp and Robert J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4365–4384. MR 1308023, DOI 10.1090/S0002-9947-1995-1308023-8
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776