Belov 1996 estimate free term nonnegative trigonometric polynomial

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corollary_1: Belov and Konyagin's bounds for the least free terms of nonnegative cosine polynomials with nonincreasing integer coefficients: (ln n)^2/ln ln n << K(n) << M(n) << (ln n)^3 for all n at least 3.

corollary_2: Belov and Konyagin's bound ln f(n) << (ln n)^4 for n at least 2, where f(n) is the least, over positive integers k_1, ..., k_n, of the maximum modulus of the product of the factors 1 - e^(itk_j).

theorem_1: Belov and Konyagin's theorem that the sequence n^q is strictly admissible for every q in (1,2], with an explicit lower bound for its sine sum, that an explicit set built from odd primes is strictly admissible for beta at least 2^14, and that no sequence with inf lambda_(n+1)/lambda_n > 1 is admissible.

theorem_2: Belov and Konyagin's two-sided comparison, for every natural n, of the least free terms M(n) and K(n) of nonnegative cosine polynomials with nonincreasing integer coefficients with a functional Phi defined by an infimum over admissible sequences, up to the constants 1/120, 11/5 and 16/5.

theorem_3: Belov and Konyagin's theorem that, for independent uniform variables xi_n on (0,1), the random sequences (n+lambda-1)^q/xi_n with q > 1 and lambda > cq^3, and nu^((n-1)^(1/3))/xi_n with nu in (1,nu_0], are strictly admissible with probability greater than 1/2.

theorem_4: Belov and Konyagin's main theorem: the admissible-sequence functional Phi satisfies (ln x)^2/ln ln x << Phi(x) << (ln x)^3 for all x at least 3.

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Belov, A. S. and Konyagin, S. V., An estimate for the free term of a nonnegative trigonometric polynomial with integer coefficients. Mat. Zametki 59 (1996), no. 4, 627--629.

This short Russian-language note (Kratkie soobshcheniya) states its results without proofs. It studies 'admissible' sequences, meaning positive reals with a convergent reciprocal sum whose associated sine series is nonnegative for all x >= 0, and applies them to the least constant term of a nonnegative cosine polynomial with integer coefficients. Theorem 1 (pp. 627-628) states that for every q in (1,2] the sequence n^q is strictly admissible, with the quantitative bound sum sin(pi x / n^q) > x^{1/q}/(20(1-1/q)) for x >= 1/2; it also exhibits an explicit strictly admissible set built from odd primes for beta >= 2^14, and shows that no positive sequence with inf lambda_{n+1}/lambda_n > 1 is admissible. Write M_Z^dec(n) for the least a_0 over positive integers a_1 >= ... >= a_n with sum_{k=0}^n a_k cos(kx) >= 0 for all x, K_Z^dec(n) for the least a_0 over nonincreasing nonnegative integers a_1, a_2, ... with sum a_k = n and the same nonnegativity, and K_Z(n) <= K_Z^dec(n) for the version without monotonicity. Theorem 2 (p. 628) compares these with an auxiliary function Phi defined as an infimum over admissible sequences, giving (1/120)Phi(n) <= M_Z^dec(n) <= (11/5)Phi(n) and (1/120)Phi(n/(7Phi(n))) <= K_Z^dec(n) <= (16/5)Phi(n) for all natural n. Combining Theorem 2 with Theorem 1(2) yields K_Z^dec(n) << M_Z^dec(n) << Phi(n) << (log n)^5 for n >= 2; the earlier bounds the note recalls are K_Z(n) = O(n^{1/3}(log n)^{1/3}) (Odlyzko), the same without the logarithmic factor (Kolountzakis), and K_Z^dec(n) <= 88 exp(sqrt(2 log n log log n)) and M_Z^dec(n) <= 8 exp(sqrt(2 log n log log n)) for n >= 3 (Belov). Theorem 3 (p. 629) shows that suitable random sequences are strictly admissible with probability greater than 1/2, and Theorem 4 (p. 629), the note's main result, gives (log x)^2/log log x << Phi(x) << (log x)^3 for x >= 3; the note says either part of Theorem 3 gives the upper bound and the lower bound develops ideas of Belov's 1981 paper. Corollary 1 (p. 629) gives (log n)^2/log log n << K_Z^dec(n) << M_Z^dec(n) << (log n)^3 for n >= 3. Through Odlyzko's inequality log f(n) < K_Z(n)(1 + log n), where f(n) is the Erdos-Szekeres minimax product, Corollary 2 (p. 629) gives log f(n) << (log n)^4 for n >= 2.

Source: https://www.mathnet.ru/eng/mzm1756. The file prints "© А. С. Белов, С. В. Конягин 1996" at the foot of p. 1 (read on the page image, the text layer being garbled) and no license wording, 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.

Bears on. #256: the note's f(n) is the problem's f(n), and Corollary 2 states log f(n) << (log n)^4 for n >= 2, so log f(n) >> n^c fails for every c > 0; the note announces the bound without proof and gives no lower bound for f(n).

Results.

• Theorem 1 (pp. 627-628): n^q is strictly admissible for q in (1,2], with a lower bound for its sine sum; a prime-based set is strictly admissible for beta >= 2^14; lacunary sequences are not admissible.
• Theorem 2 (p. 628): (1/120)Phi(n) <= M_Z^dec(n) <= (11/5)Phi(n) and (1/120)Phi(n/(7Phi(n))) <= K_Z^dec(n) <= (16/5)Phi(n) for all natural n, with the consequence K_Z^dec(n) << M_Z^dec(n) << Phi(n) << (log n)^5 for n >= 2.
• Theorem 3 (p. 629): the random sequences (n+lambda-1)^q/xi_n (q > 1, lambda > cq^3) and nu^{(n-1)^{1/3}}/xi_n (nu in (1,nu_0]) are strictly admissible with probability greater than 1/2.
• Theorem 4 (p. 629): (log x)^2/log log x << Phi(x) << (log x)^3 for x >= 3.
• Corollary 1 (p. 629): (log n)^2/log log n << K_Z^dec(n) << M_Z^dec(n) << (log n)^3 for n >= 3.
• Corollary 2 (p. 629): log f(n) << (log n)^4 for n >= 2.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.