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

up vote 25 down vote accepted

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

  • 2
    Do estimates on the constant $C$ exist? – lcv Oct 8 at 17:43
  • 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 – BigM Oct 12 at 21:01

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