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Statement

Admissible sequences are defined on the Theorem 1 page. The following notation is the note's (p. 628).

• For real $t$, $\varphi_\Lambda(t)$ is the number of terms of $\Lambda$ not exceeding $t$, and for $v>0$

$$\Phi(v)=\inf\Bigl\{v\int_v^\infty\frac{\varphi_\Lambda(t)}{t^2}\,dt: \Lambda\text{ admissible},\ \min\Lambda=1\Bigr\}.$$

The note states that $\Phi$ is positive, strictly increasing and concave on $(0,\infty)$.

• For $n\in\mathbb N$, $M^{\downarrow}_Z(n)=\min a_0$, the minimum over natural numbers $a_1,\dots,a_n$ with $a_1\ge\cdots\ge a_n$ and $\sum_{k=0}^n a_k\cos(kx)\ge0$ for all $x$.
• $K^{\downarrow}_Z(n)=\inf\alpha_0$, the infimum over nonnegative integers $\alpha_1,\alpha_2,\dots$ with $\sum_{k=1}^\infty\alpha_k=n$, $\sum_{k=0}^\infty\alpha_k\cos(kx)\ge0$ for all $x$, and $\alpha_1\ge\alpha_2\ge\cdots$. Dropping the last condition defines $K_Z(n)$, so $K_Z(n)\le K^{\downarrow}_Z(n)$.

Theorem 2 (p. 628). For every natural $n$,

$$\frac1{120}\Phi(n)\le M^{\downarrow}_Z(n)\le\frac{11}5\Phi(n),\qquad \frac1{120}\Phi\Bigl(\frac n{7\Phi(n)}\Bigr)\le K^{\downarrow}_Z(n)\le\frac{16}5\Phi(n).$$

The note then states (p. 628) that Theorem 2 and part 2 of Theorem 1 give, for $n\ge2$,

$$K^{\downarrow}_Z(n)\ll M^{\downarrow}_Z(n)\ll\Phi(n)\ll(\ln n)^5 .$$

Earlier bounds the note recalls (p. 628): Odlyzko's $K_Z(n)=O(n^{1/3}(\ln n)^{1/3})$ for $n\ge2$, Kolountzakis's removal of the logarithmic factor, and Belov's $K^{\downarrow}_Z(n)\le88\exp(\sqrt{2\ln n\ln\ln n})$ and $M^{\downarrow}_Z(n)\le8\exp(\sqrt{2\ln n\ln\ln n})$ for $n\ge3$.

Source. A. S. Belov and S. V. Konyagin, An estimate for the free term of a nonnegative trigonometric polynomial with integer coefficients (in Russian), Mat. Zametki 59 (1996), no. 4, 627--629. Theorem 2 and the notation on p. 628. The edition read is identified on the source card.

Read depth. Claims checked: the definitions and the statement were read clause by clause on the printed page. The note prints no proofs.

Proof pointer

None in the note.

Dependencies

None stated.

Bears on

• Problem 256: indirectly. With Theorem 4 it underlies Corollary 1, from which the note derives the bound on the problem's $f(n)$ in Corollary 2.