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Claim. There is an absolute constant $C>0$ such that for any distinct integers $m_1,\ldots,m_M$,

$$\int_{-\pi}^{\pi}\Bigl\lvert\sum_{j=1}^M e^{im_jx}\Bigr\rvert\,\mathrm dx\ge C\log M .$$

This is Littlewood's conjecture, the question of Problem 512 after the change of variable $x=2\pi\theta$, which only rescales the constant. It is the main result of S. V. Konyagin, On a problem of Littlewood, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 2, 243–265, 463, translated as Math. USSR-Izv. 18 (1982), no. 2, 205–225, filed as a source card at digest depth. The theorem on p. 207 of the translation proves more: for $F(x)=\sum_{j=1}^Na_je^{in_jx}$ with every $\lvert a_j\rvert\ge1$ and a positive integer $R$ such that $2^R$ divides no difference $n_j-n_l$, the $L^1$ distance from $F$ to a subspace of trigonometric polynomials is at least $C\log N$, and $\lVert F\rVert_1$ is at least that distance. The method decomposes $F$ by the dyadic averaging operators that project onto frequencies divisible by $2^r$ and uses their $L^1$ contraction.

Acceptance. The paper is a refereed journal publication, received 4 November 1980 according to the journal's record, the refereed evidence; the record gives the year and issue of the Russian original and no day, and this page is dated to the first day of that year. The site's curator, Thomas Bloom, labels the problem proved and credits the proof of Littlewood's conjecture independently to this paper and to McGehee, Pigno and Smith, whose claim page records their proof; that curator credit is the reviewed evidence. The Lean proof that the site's label refers to follows the method of McGehee, Pigno and Smith and is linked from their page; no formalization declares itself a formalization of this paper's proof.

Depends on. Nothing beyond the cited paper.