Konyagin 1994 minimum modulus random trigonometric polynomials coefficients

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theorem_1: Konyagin's theorem that for every eps > 0 the probability that a random trigonometric polynomial with n independent uniform plus-or-minus-one coefficients has minimum modulus on the circle greater than n^(-1/2+eps) tends to zero as n tends to infinity.

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Konyagin, S. V., On the minimum modulus of random trigonometric polynomials with coefficients {$\pm1$}. Mat. Zametki 56 (3) (1994), 80-101, 158. The file prints "© С.В. Конягин 1994", the author's copyright line, in the footer of its first page (read from the text layer) and no license wording on any of its 22 pages, and the hosting site's terms of use state that its materials "are fully copyrighted by Steklov Mathematical Institute, Russian Academy of Sciences, and/or by other copyright holder" and that reproduction or republication "requires written permission of the copyright holder" (https://www.mathnet.ru/php/agreement.phtml?option_lang=eng, read 2026-10-02), every other right reserved.

Konyagin studies P_n(u), the probability that the random polynomial T(x) = sum_{j<n} xi_j exp(ijx), with independent coefficients xi_j each equal to +1 or -1 with probability 1/2, satisfies min_{x in T} |T(x)| > u. Theorem 1 (p. 80) states that for every ε > 0, P(n^{-1/2+ε}) → 0 as n → ∞, so for all but a vanishing proportion of sign choices the minimum modulus is at most n^{-1/2+ε}. The introduction (p. 80) recalls Littlewood's conjecture that P(ε n^{1/2}) → 0 for every ε > 0, Kashin's proof of it in the form P(n^{1/2}(log n)^{-1/3}) → 0, and A. M. Odlyzko's unpublished result P(n^{1/3+ε}) → 0 together with his conjecture that most such polynomials satisfy min |T(x)| < n^{-1/2+ε}, which Theorem 1 proves. The proof (pp. 80--101), for 0 < ε < 1, looks at the points 2πκ/k with k a prime near n^{1-ε/(5r)}, shows by Taylor's formula that a suitable event for T and its first r-1 derivatives at such a point forces a small value nearby (Lemma 1.1, p. 82), estimates the probabilities of these events and of their pairwise intersections through characteristic functions (Sections 2 and 3, Lemmas 3 and 3', pp. 96 and 100), and concludes by the second moment method (pp. 100--101). The paper is written in Russian.

Read status: claims checked for Theorem 1, the setting and statement read clause by clause on the print; the proof outline was followed, and the estimates of Sections 2 and 3 were not checked step by step.

Source: https://www.mathnet.ru/eng/mzm2261.

Bears on.

• #525: applied with n+1 terms, Theorem 1 gives min_{|z|=1} |f(z)| ≤ (n+1)^{-1/2+ε} for all but o(2^{n+1}) of the degree n polynomials f with ±1 coefficients, for each fixed ε > 0; for 0 < ε < 1/2 this bound is below 1, so it answers the problem's first question yes, and it bounds the minimum in its second question from above. The paper gives no lower bound for the minimum.

Results.

• Theorem 1 (p. 80): For every ε > 0, P(n^{-1/2+ε}) → 0 as n → ∞, i.e. a random ±1 trigonometric polynomial with n terms has min_x |T(x)| ≤ n^{-1/2+ε} with probability tending to 1.

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