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Statement

Notation as on the theorem's page.

Corollary 1 (p. 223, quoted). "If the function $F(x)$ and the number $R\in\mathbf N$ satisfy the conditions of the theorem, then

$$I_{F,R}\ge C\ln\Bigl(\sum_{j=1}^N\exp|a_j|\Bigr),$$

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

The paper introduces it (p. 223) as the positive answer to Littlewood's further question whether $\|\sum_{j=1}^M\exp(im_jx)\|_1\ge C\ln M$ when $m_1,\ldots,m_M$ are not necessarily distinct, which it says is equivalent to $\|\sum_{j=1}^Na_j\exp(in_jx)\|_1\ge C\ln\sum_{j=1}^Na_j$ for distinct integers $n_j$ and positive integers $a_j$.

Proof pointer

Pp. 223--224. The paper says the corollary follows easily from the theorem and the bound $I_{F,R}\ge\max_j|a_j|\ge1$ ((30), p. 211), splitting on whether $\max_j|a_j|\le2\ln N$.

Read depth

Claims checked: the statement, the paper's account of Littlewood's question and the two-case derivation were read on the page images of the English translation. Nothing here is independently reviewed.

Dependencies

The theorem (p. 207).

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: with all $a_j=1$ the bound reads $I_{F,R}\ge C\ln(eN)$, which through $\|F\|_1\ge I_{F,R}$ gives the problem's inequality for a set of $N$ integers, as the theorem already does; the weighted form goes beyond the problem, which asks only about sets. The paper's statement for integers not necessarily distinct is Corollary 2.