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Statement

A sequence $\Lambda=\{\lambda_n\}_{n=1}^\infty$ of positive reals is admissible (p. 627) when $\sum_{n\ge1}1/\lambda_n<\infty$ and $\sum_{n\ge1}\sin(x/\lambda_n)\ge0$ for all $x\ge0$; it is strictly admissible when moreover $\sum_{n\ge1}\sin(x/\lambda_n)>0$ for all $x>0$. The note credits the definition to one of the authors (A. S. Belov, 1994).

Theorem 1 (pp. 627--628). The following hold.

1. For every $q\in(1,2]$ the sequence $\{n^q\}_{n=1}^\infty$ is strictly admissible. Moreover, for every $x\ge1/2$,

$$\sum_{n=1}^\infty\sin\Bigl(\frac{\pi x}{n^q}\Bigr)>\frac{x^{1/q}}{20(1-1/q)}.$$

2. For $\beta\ge2^{14}$ the sequence

$$\{1\}\cup\bigcup_{j=1}^\infty\ \bigcup_{p\le\beta j^2} \Bigl\{\frac{2^jp}{2i+1}\Bigr\}_{i=0}^{p(1+[\ln j])-1}$$

is strictly admissible, where $p$ runs over the odd primes and $[\,\cdot\,]$ is the integer part. 3. A sequence $\{\lambda_n\}_{n=1}^\infty$ of positive reals with $\inf\{\lambda_{n+1}/\lambda_n:n\ge1\}>1$ is not admissible.

Part 1 is on p. 627, parts 2 and 3 on p. 628.

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 1 on pp. 627--628. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed pages. The note is a short communication and prints no proofs.

Proof pointer

None in the note. Part 2 is what the note combines with Theorem 2 to get the bound $\ll(\ln n)^5$ displayed on p. 628.

Dependencies

None stated.

Bears on

• Problem 256: indirectly. Part 2 gives, through Theorem 2, the bound $K^{\downarrow}_Z(n)\ll(\ln n)^5$, which the note then improves to $(\ln n)^3$ in Corollary 1; the bound on the problem's $f(n)$ is Corollary 2.