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Statement

Conventions (pp. 205, 207). $\mathbf T=[-\pi,\pi)$ carries the normalized measure $d\mu=dx/2\pi$, $\int$ is taken with respect to it, and $\|f\|_1=\int|f(x)|$. The frequencies $n_1,\ldots,n_N$ are distinct integers, $\mathbf Z_+$ is the set of nonnegative integers and $\mathbf N$ the set of positive integers, $\operatorname{Sp}(f)$ is the set of $n\in\mathbf Z$ with $\hat f(n)\ne0$, and $C$ is a positive constant, not necessarily the same at each occurrence. For $F\in L(\mathbf T)$ and $r\in\mathbf Z_+$, $X=X_{F,r}$ is the subspace of $L(\mathbf T)$ spanned by the functions $\exp(i(n+2^ru)x)$ with $n\in\operatorname{Sp}(F)$ and $u\in\mathbf N$, and (5)

$$I_{F,r}=\inf_{F_r\in X}\|F-F_r\|_1 .$$

Since $X_{F,0}\supset X_{F,1}\supset\cdots$, the quantities $I_{F,r}$ are nondecreasing in $r$ (p. 207).

Theorem (p. 207). Let $F(x)=\sum_{j=1}^Na_j\exp(in_jx)$ with $a_j\in\mathbf C$ and $|a_j|\ge1$ for $j=1,\ldots,N$ (6), and let $R$ be a positive integer such that $2^R\nmid(n_j-n_l)$ for $1\le j<l\le N$ (7). Then (8)

$$I_{F,R}\ge C\ln N,$$

where $C>0$ is an absolute constant.

Since $\|F\|_1\ge I_{F,R}$, the paper records (p. 207) that the theorem contains the inequality $\|F\|_1\ge C\ln N$. For distinct frequencies some $R$ satisfying (7) always exists, so this lower bound holds for every such $F$; with all $a_j=1$ it is Littlewood's conjecture, stated in the paper as Corollary 2.

Proof pointer

Pp. 207--223. The proof works with the dyadic averages $f_r$ of $f$, whose Fourier series keep the frequencies divisible by $2^r$, and their differences $\tilde f_r=f_r-f_{r+1}$, which keep those divisible by $2^r$ but not by $2^{r+1}$ ((10), (11), p. 207); both have $L^1$ norm at most $\|f\|_1$ ((9), p. 207). The printed definition of $f_r$ omits the factor $2^{-r}$ that the formula for $\tilde f_r$ and (9)--(11) require. Lemma 1 (p. 208) is a duality lower bound for $\|f\|_1$ in terms of $I_{f,r}$ and a bounded test function supported on frequencies divisible by $2^r$; with it Lemma 3 (p. 211) gives $I_{F,R}\ge\#(A_\Gamma)^{1/3}/5$, where $A_\Gamma$ is the set of $r\in\mathbf Z_+$ such that some $n_j$ is divisible by $2^r$ but not by $2^{r+1}$ (p. 208), and Lemma 4 (p. 212) gives $I_{F,R}\ge\ln N/(8\ln\ln N)$ for $N\ge4$. Lemma 5 (pp. 213--214), which the paper calls a generalization and refinement of Lemma 5 of Pichorides (its reference [15], p. 208), is the second recursion inequality. Section 5 (pp. 218--223) proves by induction on $N$ the stronger bound $I_{F,R}\ge C\ln N+\ln N/(10\ln\ln N)$ (62): Lemma 8 (p. 218) splits into the case that many dyadic levels are occupied, handled by Lemma 3, and the case of a chain of levels $r_0<\cdots<r_{2n}$ each carrying many frequencies, handled by the induction hypothesis and the recursion inequalities.

Read depth

Claims checked: the conventions, the definition (5) and the theorem were read clause by clause on the page images of the English translation; the structure of the proof was followed but its estimates were not checked. Nothing here is independently reviewed.

Dependencies

None in the corpus. The paper cites a duality theorem (its reference [16], p. 142), facts on Hardy spaces from Zygmund's Trigonometric series and a lemma of Pichorides (its reference [15]).

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: the inequality $\|F\|_1\ge C\ln N$ that the theorem contains, taken with all $a_j=1$, is the problem's inequality for a set of $N$ integers: with the normalized measure, $\|F\|_1$ equals the problem's integral after the change of variable $x=2\pi\theta$.