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Statement

Here $\|f\|_1=\int|f(x)|$ with the normalized measure $dx/2\pi$ on $\mathbf T=[-\pi,\pi)$ (p. 205).

Corollary 2 (p. 224, quoted). "For any integers $m_1,\ldots,m_M$ (not necessarily distinct)

$$\Bigl\|\sum_{j=1}^M\exp(im_jx)\Bigr\|_1\ge C\ln M,$$

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

The abstract (p. 205) states the same inequality for any integers $m_1,\ldots,m_M$ with the integral $\int_{-\pi}^{\pi}\cdots\,dx$ written out, as the proof of the Littlewood conjecture.

Proof pointer

P. 224. The paper prints no separate proof. Grouping equal frequencies writes the sum as $\sum_ja_j\exp(in_jx)$ with distinct $n_j$ and positive integer $a_j$ summing to $M$, which is the form of Littlewood's question the paper says Corollary 1 answers (p. 223).

Read depth

Claims checked: the statement and the abstract were read on the page images of the English translation. Nothing here is independently reviewed.

Dependencies

Corollary 1 and through it the theorem.

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 512: for distinct $m_j$ forming a set $A$ of size $M$ this is the problem's inequality: with the normalized measure, the norm equals the problem's integral after the change of variable $x=2\pi\theta$.