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Statement

Corollary 3 (p. 224, quoted). "If $n_1,\ldots,n_N$ are distinct positive integers, $\varphi_j$ are real numbers, and $a_j$ are real numbers satisfying the condition $|a_j|\ge1$ ($j=1,\ldots,N$), then

$$\Bigl|\min_{x\in\mathbf T}\sum_{j=1}^Na_j\cos(n_jx+\varphi_j)\Bigr|\ge C\ln\sum_{j=1}^N\exp|a_j|\ge C\ln N,$$

where $C>0$ is an absolute constant."

In particular (p. 224), with

$$H_N=\inf_{\{n_1,\ldots,n_N\},\ n_1>\cdots>n_N>0}\Bigl|\min_{x\in\mathbf T}\sum_{j=1}^N\cos(n_jx)\Bigr|,$$

$H_N\ge C\ln N$. The paper calls the estimation of $H_N$ the problem of Ankeny and Chowla and remarks that its bound differs strongly from the known upper estimate, of order $N^{1/2}$.

Proof pointer

P. 224. The paper splits the sum $f$ into its positive and negative parts $f^+$ and $f^-$, notes $\|f^+\|_1=\|f^-\|_1=\|f\|_1/2$ since $f$ has mean zero, bounds $|\min f|\ge\|f^-\|_1$ and applies Corollary 1.

Read depth

Claims checked: the statement, the definition of $H_N$ and the derivation were read on the page images of the English translation. Nothing here is independently reviewed.

Dependencies

Corollary 1.

Source. S. V. Konyagin, On the Littlewood problem, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 2, 243--265, 463; English translation, On a problem of Littlewood, Math. USSR Izvestija 18 (1982), no. 2, 205--225, whose pages are cited here; the edition read is named on the source card.

Bears on

• Problem 510: the bound $H_N\ge C\ln N$ is a lower bound of logarithmic order for the size of the minimum of a cosine sum over $N$ distinct positive integers, where the problem asks for order $N^{1/2}$; it does not decide the problem.