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The following is a conjecture due to Littlewood.

For any set of distinct non-zero integers $n_1,\ldots,n_k$ the inequality $$\int_0^{2\pi}|1+e^{in_1x}+\cdots+e^{in_kx}| \, dx\geq C\log k$$ holds.

Has this proven to be true or false?

Update 1. An extension to finite fields can be found here

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This was proved by S. Konyagin [7] and independently by McGehee, Pigno, and Smith [13] in 1981. A short proof is available in [5].

[7] S.V. Konjagin, On a problem of Littlewood, Mathematics of the USSR, Izvestia, 18 (1981), 205–225. http://mi.mathnet.ru/eng/izv1556

[5] R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, 1993.

[13] O.C. McGehee, L. Pigno, and B. Smith, Hardy’s inequality and the L1 norm of exponential sums, Ann. Math. 113 (1981), 613–618. https://www.jstor.org/stable/2007000

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    $\begingroup$ Do estimates on the constant $C$ exist? $\endgroup$
    – lcv
    Commented Oct 8, 2018 at 17:43
  • $\begingroup$ Glancing at Konyagin's paper( it's in Russian and I worked so hard to follow his proof) its stated that $C\leq 1$. But again its been over three decades and I'm pretty sure better estimates are out there $\endgroup$
    – BigM
    Commented Oct 12, 2018 at 21:01
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    $\begingroup$ I'm 4 years late, but see Stegeman, Math Annalen 261, 51-54 (1982) link.springer.com/article/10.1007/BF01456409. It is natural to conjecture you can take $C = 4/\pi^2$ (which is what you get from arithmetic progressions), and Stegeman shows $C = 4/\pi^3$ is acceptable. $\endgroup$
    – Sean Eberhard
    Commented Oct 11, 2022 at 9:12
  • $\begingroup$ In fact Stegeman does a little better, giving a value of $C=0.1293...$ (obtained as the solution to a complicated transcendental equation). Stegeman gives $4/\pi^3=0.1290...$ as the headline result presumably because it's the nearest 'nice-looking constant' and resembles the conjectured $4/\pi^2$. $\endgroup$
    – Thomas Bloom
    Commented Oct 14, 2022 at 21:57

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