Konyagin 1981 littlewood problem

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corollary_1: Konyagin's weighted form of his theorem: under the theorem's hypotheses I_{F,R} is at least C ln of the sum over j of exp|a_j|, which answers Littlewood's question for not necessarily distinct frequencies.

corollary_2: Konyagin's statement of the Littlewood conjecture: for any integers m_1, ..., m_M, not necessarily distinct, the L1 norm of the sum of exp(i m_j x) is at least C ln M with C an absolute constant.

corollary_3: Konyagin's bound for real trigonometric polynomials: for N distinct positive integers, phases phi_j and real a_j with |a_j| >= 1, the modulus of the minimum of the sum of a_j cos(n_j x + phi_j) is at least C ln N, a logarithmic lower bound in the Ankeny-Chowla cosine problem.

theorem: Konyagin's main theorem: for a sum of N exponentials with distinct frequencies and coefficients of modulus at least 1, and R such that 2^R divides no difference of frequencies, the distance I_{F,R} is at least C ln N, hence the L1 norm of the sum is at least C ln N.

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Konyagin, S. V., On the Littlewood problem. Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 2, 243-265, 463. The copy read for this card is the American Mathematical Society's English translation (Math. USSR Izvestija 18 (1982), no. 2, 205-225), not the Russian original cited above, and it prints "©1982 American Mathematical Society" in its first-page footer with no license wording, every other right reserved.

The paper (read in the English translation) proves the Littlewood conjecture. Its abstract (p. 205) states that for any integers $m_1,\ldots,m_M$ the integral over $[-\pi,\pi]$ of $|\sum_{j=1}^M\exp(im_jx)|$ is at least $C\ln M$ for an absolute constant $C>0$, and Corollary 2 (p. 224) restates this for integers not necessarily distinct. Norms are taken with the normalized measure $dx/2\pi$ on $\mathbf T$ (p. 205). Section 1 reviews the earlier partial results the theorem supersedes, including Cohen's $\|\sum_{j=1}^N\exp(in_jx)\|_1\ge C(\ln N/\ln\ln N)^{1/8}$ ($N\ge3$) and Pichorides's $\|f\|_1\ge C\ln N/(\ln\ln N)^2$ for coefficients with $|a_j|\ge1$ and $N\ge3$ (pp. 206--207).

The main theorem (p. 207) proves more than the conjecture: for $F(x)=\sum_{j=1}^Na_j\exp(in_jx)$ with distinct $n_j$, all $|a_j|\ge1$, and a positive integer $R$ such that $2^R$ divides no difference $n_j-n_l$ with $j<l$, the distance $I_{F,R}=\inf_{F_R\in X}\|F-F_R\|_1$ from $F$ to the subspace $X$ of $L(\mathbf T)$ spanned by the functions $\exp(i(n+2^Ru)x)$ with $n\in\operatorname{Sp}(F)$ and $u\in\mathbf N$ is at least $C\ln N$; since $\|F\|_1\ge I_{F,R}$, the Littlewood inequality follows. The method decomposes $f$ by the dyadic averaging operators $f_r(x)=\sum_{j=0}^{2^r-1}f(x+\pi j/2^{r-1})$ (as printed; the normalizing factor $2^{-r}$ that the formula for $\tilde f_r$ and (9)--(11) require is missing from the printed definition) and their differences $\tilde f_r=f_r-f_{r+1}$, whose Fourier series keep the frequencies divisible by $2^r$, respectively divisible by $2^r$ but not by $2^{r+1}$, and uses $\|f\|_1\ge\|f_r\|_1$ and $\|f\|_1\ge\|\tilde f_r\|_1$ (p. 207), combined in a first recursion inequality (Section 2) and a second one (Section 4). Section 6 (pp. 223--224) derives a weighted form, $I_{F,R}\ge C\ln\sum_j\exp|a_j|$ (Corollary 1), and a lower bound $C\ln N$ for the modulus of the minimum of a real cosine sum over $N$ distinct positive integers (Corollary 3), which the paper relates to the cosine problem of Ankeny and Chowla.

Read status: claims checked for the theorem and Corollaries 1 to 3, on the page images of the English translation; the proof's estimates were not checked. Result pages: theorem, corollary_1, corollary_2 and corollary_3.

Source: https://www.mathnet.ru/eng/im1556.

Bears on. #512: Corollary 2 (p. 224), which for distinct frequencies is the case $a_j=1$ of the theorem (p. 207), gives the problem's inequality for every finite set of $N$ integers: with the normalized measure the norm equals the problem's integral after the change of variable $x=2\pi\theta$. #510: Corollary 3 (p. 224) bounds the modulus of the minimum of $\sum_{j=1}^N\cos(n_jx)$ below by $C\ln N$ for distinct positive integers $n_j$, where the problem asks for order $N^{1/2}$; the paper notes the known upper estimate has order $N^{1/2}$. It does not decide the problem.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.