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?

up vote 23 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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