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](https://www.researchgate.net/profile/Victor_Garcia11/publication/327356917_The_finite_Littlewood_problem_in_mathbb_F_pFp/links/5b969d5792851c78c410183b/The-finite-Littlewood-problem-in-mathbb-F-pFp.pdf?origin=publication_detail)



A conjecture of Littlewood
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>


